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The production of hypersonic shock waves in an electrothermal diaphragm shock tube Phillips, Malvern Gordon Rutherford 1969

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\\7b THE PRODUCTION OF HYPERSONIC SHOCK WAVES IN AN ELECTROTHERMAL DIAPHRAGM SHOCK TUBE by MALVERN GORDON RUTHERFORD PHILLIPS B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT The operation of a diaphragm shock tube of 5 cm inside diameter i n which the driver gas i s heated by the discharge of e l e c t r i c a l energy ( ~ 10 joules) i s analyzed i n d e t a i l . A technique is described for the measurement of the heated driver gas pressure and empirical r e l a t i o n s are obtained which enable the shock speed to be calculated from a knowledge of the di s -charge voltage and test gas pressure. Using helium dr i v e r gas i n i t i a l l y at atmospheric pressure, shock Mach numbers of about 20 are obtained in argon at an i n i t i a l pressure of about 1 Torr. The separation of shock front and contact surface i s analyzed by means of a convenient shock r e f l e c t i o n technique using a smear camera. The properties of the. shock-heated gas are shown to agree with the predictions of standard shock wave theory, which y i e l d s a temperature of about 1.3*10^°K and an 17 -3 electron density of about IQ cm for the case of a Mach 20 shock i n argon at 0.5 Torr. In th i s case the shock-heated gas sample i s observed to be about 5 cm i n length at a p o s i t i o n 1.2 meters from the diaphragm. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS i x CHAPTER.l' INTRODUCTION 1.1 I n s t r u c t i o n s to the Reader 1 1.2 The B a s i c Problem 1 1.3 O u t l i n e of the Thesis 6 CHAPTER 2 SHOCK WAVES AND SHOCK TUBES 2.1 I n t r o d u c t i o n 9 2.2 Shock Waves i n I d e a l Gases 9 2.3 Shock Waves i n Real Monatomic Gases 13 2.4 Dependence of Shock Speed on Diaphragm 15 Pressure R a t i o 2.5 S o l u t i o n of the Equations 19 2.6 The Shock-Heated Gas 25 2.7 Boundary Layers 2 8 2.8 The E f f e c t of Shock Tube E x t r e m i t i e s 29 CHAPTER 3 APPARATUS 3.1 I n t r o d u c t i o n 31 3.2 The D r i v e r S e c t i o n 31 3.3 The Dischai-ge C i r c u i t 34 3.4 The Diaphragm 34 i v 3.5 The Test S e c t i o n ' 38 3.6 Instrumentation 41 3.7 E l e c t r i c a l Measurements 43 CHAPTER 4 THE DRIVER 4.1 I n t r o d u c t i o n 45 4.2 The D r i v i n g Mechanism 45 4.3 Determination of the D r i v e r Gas Pressure 49 CHAPTER 5 COMPARISON OF SHOCK TUBE OPERATION WITH IDEAL THEORY 5.1 I n t r o d u c t i o n 62 5.2 Dependence of Shock Speed on A x i a l P o s i t i o n 63 5.3 Comparison of Shock Tube Operation with 67 Ideal Theory 5.4 The E f f i c i e n c y of the Shock Tube 77 CHAPTER 6 THE SHOCK-HEATED GAS 6.1 I n t r o d u c t i o n 81 6.2 Extent of the Region of Shock-Heated Gas 83 6.3 Currents i n the Shock-Heated Gas 105 6.4 P r o p e r t i e s of the Shock-Heated Gas 108 CHAPTER 7 CONCLUSIONS 7.1 Conclusions 119 7.2 Suggestions f o r Future Work 122 BIBLIOGRAPHY 1 124 APPENDIX - PRESSURE PROBES 127 V L I S T OF TABLES Page T a b l e I S h o c k Tube D r i v e r C o n d i t i o n s 72 T a b l e I I E l e c t r i c a l E f f i c i e n c y o f t h e S h o c k Tube 79 T a b l e I I I T h e r m a l E f f i c i e n c y o f t h e S h o c k Tube 80 v i LIST OF FIGURES Page 1-1 Pressure-Driven Shock Tube 3 1- 2 Comparison of Types of Shock Tubes 7 2- 1 Shock Wave i n Laboratory and Shock-Fixed 10 Frames of Reference 2-2 Gas Flow i n Shock Tube A f t e r Diaphragm Bursts 18 2-3 Pressure Ratio across Shocks i n Argon 20 2-4 Density and V e l o c i t y Ratios across Shocks i n 21 Argon 2-5 Temperature Behind Shocks i n Argon 22 2-6 Degree of I o n i z a t i o n (a) i n Shock-Heated Argon 23 2-7 Shock Mach Number (M) as. a Function of Diaphragm 24 Pressure Ratio (p^/p-^) f° r Shocks i n Argon Driven by Helium 2- 8 Luminosity S t r u c t u r e i n Strong Shock .27 3- 1 Schematic Diagram of V e r t i c a l S e c t i o n through 32 Shock Tube Axis 3-2 D r i v e r Section of Shock Tube 33 3-3 D e t a i l s of Discharge C i r c u i t 35 3-4 Two Views of the D r i v e r Section 37 3-5 Diaphragms Before and A f t e r B u r s t i n g 37 3-6 H o r i z o n t a l A x i a l S e c t i o n through S p e c i a l 40 Test S e c t i o n 3- 7 S i m p l i f i e d Diagram of Pressure Probe and Housing 42 4- 1 Shock Speeds With and Without Backstrap 47 4-2 Dependence of Shock Speed on Discharge Current 48 For D r i v e r w i t h Backstrap 4-3 Diaphragm P e t a l Geometry 50 V l l . 4-4 Arrangement f o r Observing Diaphragm Opening 51 4-5 Smear Photograph of Diaphragm Opening 51 4-6 Dependence of Diaphragm Opening Time on 53 Bank Voltage 4-7 Dependence of Diaphragm Opening Time on 54 , v : Diaphragm Thickness 4-8 Arrangement f o r Observing Diaphragm Opening 56 w i t h Cold P r e s s u r i z e d D r i v e r 4-9 P h o t o m u l t i p l i e r and Pressure Transducer 57 Records of Diaphragm Opening 4-10 Diaphragm Dynamics f o r Cold D r i v e r Gas 57 4-11 Current Waveform and Diaphragm Rupture Time 60 4- 12 Times C h a r a c t e r i s t i c of Discharge and Diaphragm 60 5- 1 Dependence of Shock Speed on Distance from 64-65 the Diaphragm 5-2 Dependence of Shock Speed on P o s i t i o n f o r 69 D i f f e r e n t Diaphragm Thicknesses 5-3 Dependence of L on the Product v • t. 70 1 max op 5-4 Dependence of Shock Speed on P o s i t i o n f o r 71 D i f f e r e n t D r i v e r C o n d i t i o n s 5-5 Measured Shock Speed versus log-j^Cp^/p,)(V c=14KV) 73 5-6 Measured Shock Speed versus log-j^Cp^/p,)(V c=10KV) 74 5-7 Measured Shock Speed versus l o g 1 0 ( p ^ / P ] ^ (V = 7KV) 75 5- 8 I n i t i a l D r i v e r Gas Flow P a t t e r n 76 6- 1 A x i a l S e c t i o n of Shock Tube Showing Regions of Flow 82 6-2 Smear P i c t u r e of Shock Taken w i t h S l i t 82 P e r p e n d i c u l a r to Shock Tube Axis 6-3 Smear P i c t u r e s of Gas Flow i n Shock Tube 87-89 6-4 E x p l a n a t i o n of the Features of the Smear Camera 90 Photographs of Fi g u r e 6-3 6-5 Dependence of Shock Mach Number (M) and Shock- 92 Heated Gas Thickness (1 ) on T e s t Gas Pressure (p,) v i i i 6-6 Comparison of Measured Thickness of Shock- 93 Heated Gas (1 ) w i t h Ideal Value (1., ) m v th/ 6-7 Contact Surface Speed as a Function of Shock Speed 95 6-8 Comparison of Density Ratios 96 6-9 Experimental Check on the V a l i d i t y of 100 Equation (6.10) 6-10 Two S i t u a t i o n s i n which the I n t e g r a l of Equation 101 (6.8) may be Approximated as i n Equation (6.10) 6-11 Attempts t o Observe Currents i n Shock-Heated Gas 107 6-12 Dependence of Pressure Probe S i g n a l Amplitude 110 on P lV s 2 6-13 D e t a i l s of Spark Gap Used to Produce "Sound" Pulse 112 6-14 Arrangement f o r Measuring Sound Speed i n Shock- 114 Heated Gas 6-15 Determination of Sound Speed i n Shock-Heated Gas 116 A - l B a s i c Features of P i e z o e l e c t r i c Pressure Probe 127 A-2 C o n s t r u c t i o n D e t a i l s of Pressure Probe Housing 132 A-3 T y p i c a l Pressure Probe"Signals 133 ix ACKNOWLEDGEMENTS I wish to thank Dr. F. L. Curzon for his patient supervision and encouragement during the course of t h i s research. The suggestions of Dr. B. Ahlborn, Dr. A. J . Barnard, and Dr. J . Meyer i n the preparation of the thesis are very much appreciated. I am also indebted to a l l members of the Plasma Physics Group, p a r t i c u l a r l y Dr. B. Ahlborn, Mr. J . -P. Huni, and Mr. J . D. Strachan, for h e l p f u l discussions. I am g r a t e f u l to Mr. Huni for the loan of the smear camera. The technical assistance of Mr. A. Fraser and his s t a f f - i n p a r t i c u l a r , Mr. P. Haas, Mr. D. Stonebridge, and Mr. T. Knopp, who constructed various parts of the shock tube - i s g r a t e f u l l y acknowledged. My thanks are also due to Mr. R. Haines, for his help i n the student shop, to Mr. J . Lees and Mr. E. Williams, for t h e i r advice and help with the glassware, and to Mr. J . Dooyeweerd, Mr. D. Sieberg, Mr. J . Aazam, Mr. R. Da Costa, and Mr. W. R a t z l a f f , for cheerful assistance i n the maintenance of the equipment. F i n a l l y , I am very g r a t e f u l to the National Research Council of Canada for f i n a n c i a l assistance throughout the course of the work. CHAPTER 1 INTRODUCTION 1.1 Instructions to the Reader This thesis deals with the laboratory production of strong shock waves by e l e c t r i c a l l y heating the driver gas in a diaphragm shock tube. Section 1.2 sets out the basic concepts relevant to present shock tube technology, discusses the lim i t a t i o n s on conventional shock tubes, and describes a new shock tube (invented by P. R. Smy^5'6^) which is the subject of th i s work. Readers thoroughly f a m i l i a r with the f i e l d are referred f i r s t to Section 1.3, for an outline of the the s i s , and then to Chapter 7, where the conclusions and contributions are summarized. 1.2 The Basic Problem When a shock wave passes through a gas, the gas i s com-pressed and heated. From a knowledge of the shock speed, V and the i n i t i a l gas pressure, density, and temperature (p,, p,, and T,, r e s p e c t i v e l y ) , the properties of the compressed gas (P2> P £ » "^ 2^  c a n ^ e c a l c u l a - t e d » i d e a l l y at l e a s t , with good accuracy. I f the shock speed i s s u f f i c i e n t l y high, the heated gas may be ionized. The a b i l i t y of shock waves to produce hot gas samples with predictable properties has i n recent years aroused the int e r e s t of s c i e n t i s t s and engineers. P a r t i c u l a r l y , i n the general f i e l d of plasma physics, shock waves have been and continue to be very useful to o l s . It is important to check, for example, theories for plasma transport properties under known conditions. Accurate values for atomic t r a n s i t i o n p r o b a b i l i t i e s are necessary in astro-physical studies as well as i n the area of spectroscopic diagnosis of laboratory plasmas. Shock tubes have proved useful i n determining these quantities. Theories for spectral l i n e broadening and s h i f t are currently being complemented by data obtained from shock wave experiments. The mechanisms through which i o n i z a t i o n occurs, though not yet well understood, have been p a r t i a l l y elucidated i n work with shock waves. In the area of magnetohydrodynamics, shock wave plasmas are very useful because the plasma can be produced i n the absence of magnetic and e l e c t r i c f i e l d s . The experimentalist i s therefore at l i b e r t y to impose f i e l d s of his choice. Two techniques for producing shock waves i n the laboratory have become conventional. Each has advantageous aspects and l i m i t a t i o n s . The pressure driven shock tube, shown schematically in Figure 1-1, consists of a long tube divided into two sections by a thi n membrane. The driver section i s f i l l e d with gas at a high pressure, p^, while the test section contains the gas to be studied at a suitable pressure, p^. When the membrane i s broken, driver gas rushes into the test gas at high speed, V j . A shock wave then develops and propagates ahead of the driver gas "piston" at a speed, *V"s, while the compressed test gas c o l l e c t s between the shock front and the leading edge of the driver gas, which i s c a l l e d the contact surface (Figure 1-lb). The features of the gas flow i n the shock tube are 3 D r i v e r Gas Test Gas Pressure P4 Pressure p^ D r i v e r S e c t i o n Diaphragm Test S e c t i o n (a) Before Diaphragm Bursts Compressed Test Gas Test Gas J Expanded 1 D r i v e r l Gas 1 1 • Contact Surface Shock Front (b) Time t ^ A f t e r Diaphragm Bursts t £ (c) x-t Diagram x Figure 1-1 Pressure-Driven Shock Tube conveniently represented i n an x-t diagram (Figure 1-lc) . As the shock proceeds along the tube, the sample of shock-heated gas increases i n length. Shock tubes of this type have the favorable attributes of producing shock waves of almost constant speed and r e l a t i v e l y large samples of hot gas with predictable properties. Also, any combination of test and driver gases may be used. However, the shock waves cannot be launched reproducibly i n time due to the mechanical uncertainties associated with the deformation and rupture of the diaphragm. Furthermore, the application of this type of shock tube i s severely limited by the fact that even for an i n f i n i t e i n i t i a l pressure r a t i o , p^/p-^, the shock Mach number (ra t i o of shock speed to sound speed i n the cold test gas) cannot exceed where ^ and tfjj. are the adiabatic exponents of test and driver gases, respectively, while a^ and a^ are the sound speeds. For helium driver gas and argon test gas, for example, the maximum Mach number i s about 12.5 but technical r e s t r i c t i o n s on V^/]?i p r a c t i c a l l y l i m i t this value to about 10, r e s u l t i n g i n only about 1% i o n i z a t i o n . To achieve a higher degree of i o n i z a t i o n , f a s t e r shock waves are required. For th i s purpose, equation (1.1) suggests that, for a given test gas, one should use a driver gas of high sound speed, a^. The sound speed of a gas i s proper -1/2 t i o n a l to (T/m) ' , where T i s the absolute temperature of the gas, and m i s i t s molecular weight. Because of i t s low m, hydrogen driver gas provides the fast e s t shocks. However, i t s 5 high combustibility requires that s t r i c t safety precautions be observed, p a r t i c u l a r l y i f an attempt i s to be made to increase a^ by r a i s i n g the driver gas temperature. Helium i s the next best choice, from the point of view of molecular weight, and, because i t i s i n e r t , presents no hazard when heated. The driver gas sound speed can be increased most e f f i c i e n t l y by heating the gas with an e l e c t r i c a l discharge. f53") Early e f f o r t s to e l e c t r i c a l l y heat the driver gas^ J employed a p a i r of electrodes at one end of a tube i n i t i a l l y at a uniform pressure, p,. Discharge of a capacitor bank through the gas between the electrodes creates a volume of hot gas which expands to drive a shock wave. It was quickly r e a l i z e d that the shock speeds could be. further increased by electro -magnetically projecting the hot driver gas into the cold test gas To provide the electromagnetic force, the current leads are arranged i n the "backstrap" configuration (Figure l-2b) . Then, the i n t e r a c t i o n between the gaseous discharge current and the magnetic f i e l d from the backstrap current results i n a strong J x B force propelling the driver gas down the tube. Shock Mach numbers i n excess of 100 can be achieved i n this way, and-these shock tubes can be triggered reproducibly. However, two very serious l i m i t a t i o n s a r i s e : the shock rapi d l y decelerates, so that experiments must be performed close to the driver where only a small amount of shock-heated gas has accumulated behind the shock front; the properties of the shock-heated gas cannot always be accurately predicted from the shock speed since the discharge current often p e r s i s t s i n 6 the shock-heated test gas and since turbulent mixing of test and driver gases i s often so complete that no separated shock fx 4 1 front i s observed^ * 1 . Smy^combined both types of shock tubes by employing an electromagnetic driver arrangement i n a diaphragm shock tube with the hope that the advantages of both types would be combined (Figure l-2c) . High Mach numbers and low attenuation were indeed o b t a i n e d ^ . Also, the shock tube can be conveniently triggered. However, i t i s not a p r i o r i clear that such a hybrid shock tube i n h e r i t s a l l the fine q u a l i t i e s of both parents and none of th e i r defects. From an operational point of view, there i s one a l l -important question: Is a sizeable volume of shock-heated gas produced, and can i t s properties be predicted from the measured shock speed? Several subsidiary questions also come to mind: What i s the mechanism responsible for producing the high speed shocks? What role does the-diaphragm play? Can the shock speed be predicted from a knowledge of the e l e c t r i c a l energy stored i n the capacitor bank and from the i n i t i a l pressures i n the test and driver sections? Some of these questions are i n t e r - r e l a t e d but a l l must be answered i f a thorough understanding of the shock tube i s to be obtained. It was the purpose of the work reported i n this thesis to seek answers to these questions with the hope of shedding l i g h t on the flow processes occurring in the Smy shock tube. 1.3 Outline of the Thesis In Chapter 2 we review the standard shock tube theory 7 D r i v e r Gas Pressure P4 Test Gas Pressure pi D r i v e r Diaphragm Se c t i o n Test Section (a) Conventional Pressure-Driven Shock Tube Backstrap FO ^Current. HP Net Force '^Backstrap Magnetic F i e l d Test Gas Pressure p Capacitor (b) E l e c t r o m a g n e t i c a l l y Driven Shock Tube D r i v e r Gas I n i t i a l l y at Atmospheric Pressure E l e c t r o d e s Test Gas Pressure p Diaphragm HH Capacitor (c) Sir.y's Shock Tube Figur e 1-2 Comparison of Types of Shock Tubes pertinent to our experiments. In particular,, the equations r e l a t i n g the shock-heated gas conditions to the i n i t i a l test gas conditions and the shock speed are discussed, as well as the dependence of the shock speed on the i n i t i a l driver and test gas conditions i n a conventional diaphragm shock tube. We present results of calculations for the i n t e r e s t i n g case of shocks i n argon driven by helium. The construction of the Smy shock tube is discussed i n Chapter 3 together with the basic instrumentation employed i n the in v e s t i g a t i o n . Some improvements which we have made to the design of the shock tube are noted i n Sections 3.2 and 3.4. Our investigation of the driving mechanism responsible for the production of the high speed shocks i s presented i n Chapter 4 where the dominance of the electrothermal driver heating over electromagnetic forces i s established. In Section 4.3, a technique which we have developed to measure the dri v e r pressure i s described. An empirical r e l a t i o n for pre-d i c t i n g the shock speed from the known i n i t i a l conditions was found for this tube and i s discussed i n Chapter 5. In Chapter 6, the shock-heated gas i t s e l f i s examined. We applied a r e f l e c t e d shock technique to investigate the duration of uniform flow and the mixing of test and driver gases. The thermodynamic properties of the shock-heated gas were studied by means of pressure and sound speed measurements. In summary, we have used several experimental techniques and t h e o r e t i c a l models not previously applied to investigations of this nature to e s t a b l i s h that high speed shock waves and well-behaved plasmas can be produced with the Smy shock tube. 9 CHAPTER 2 SHOCK WAVES AND SHOCK TUBES 2.1 Introduction The purpose of thi s chapter i s to present the equations r e l a t i n g the state of the shock heated gas to the shock speed and the i n i t i a l test gas conditions. The dependence of shock speed on the diaphragm pressure r a t i o , p^/p^, i n a conventional shock tube i s also discussed. The treatment i s p a r t i c u l a r i z e d to the case of shocks i n argon driven by helium. The c l a s s i c a l equations r e l a t i n g the variables on opposite sides of the shock front are discussed by many authors (e.g. references 7 and 8 ) . Therefore t h e i r derivation w i l l not be given i n d e t a i l here. 2.2 Shock Waves i n Ideal Gases Consider a shock wave moving to the right at speed V g into gas at rest (Figure 2-la). Properties of thi s gas w i l l be denoted by a subscript 1, while properties of the gas behind the shock front, the shock heated gas, w i l l be denoted by a subscript 2. We s h a l l assume that a steady state exists and that through-out each region the gas i s i n a state of thermodynamic equilibrium. The gas flow i s assumed to be one-dimensional so that boundary layers formed at the walls of the.vessel confining the gas are ignored.. We also neglect energy losses through conduction, r a d i a t i o n , and convection. In a frame of reference i n which the shock wave i s at rest (Figure 2-lb) , the flow v e l o c i t i e s are denoted by the symbol u. The general conservation laws for mass, momentum, and energy require that 10 P2» h ' T2 -fr v, P i , y. , TX •> V Shock-Heated Test Gas Shock Front Undisturbed Test Gas (a) Laboratory Frame of Reference P2, ?a » T 2 ~ ~ • Px. f i > T l u2 = V2 " V s u l = " V s Shock-Heated Shock Test Gas Front Undisturbed Test Gas (b) Shock-Fixed Frame cf Reference hock Wave i n Laboratory and Shock-Fixed Frames of Reference J^U, = P z ^ (2.1) ft + U^,7- = + fX (2.2) Vi, -K-iu,1 = h> + i"Uj (2.3) where j>, -p, and h denote mass density, pressure, and s p e c i f i c enthalpy, respectively. Enthalpy is defined by • • . h = e + ' (2.4) where e i s the inte r n a l energy per unit mass. Equations (2.1) to (2.3) r e l a t e the gas conditions behind the shock to those i n front of the shock. They are sometimes referred to as the Rankine-Hugoniot equations. These equations may be written i n a laboratory-fixed frame of reference by the transformation u l = " V s (2.5) u 2 - v 2 - V s (2.6) where v 2 denotes the gas flow speed i n the laboratory. In order to determine the properties of region 2, the equation of state of the gas must be known as well as the dependence of e on the other v a r i a b l e s . For a perfect gas f ~ f T ( T (2.7) ' e = i KL - 1 ± (2.8) where T i s absolute temperature, R is the gas constant per unit mass, k i s Boltzmann's constant and m i s the mass of a gas p a r t i c l e . For a gas with a constant s p e c i f i c heat r a t i o , $ , the enthalpy may be expressed as Equations (2.1) to (2.3) then y i e l d A - Ji- - (*>-') ft + U + Ok. ( 2 . 1 0 ) 12 and ^ - " ^ ' - ' ^ (2 11) ft ~ (er. + op, - U . - O P , ( 2 , 1 1 } for the density r a t i o and pressure r a t i o across the shock. It i s customary to introduce the Mach number of the shock, M » Vs/a1 (2.12) where a^ i s the sound speed i n the undisturbed gas of region 1. For a perfect gas One then obtains ?r - *, +1 < 2 - 1 4 ) • j i - V U . M * - y , + i)(Ma(^.+i) +0 (2.i6) A shock i s referred to as strong i f M (or equivalently, P2/P1) is large. It is in t e r e s t i n g to note that as the shock strength increases, the density r a t i o approaches a l i m i t i n g value pf ( tfi + l ) / ( - 1) for a perfect gas. It should be pointed out that the v a l i d i t y of equations (2.1) to (2.3) does not depend on the in t e r n a l structure of the shock wave provided that this structure i s independent of time. The shock wave i n fact consists of a t r a n s i t i o n zone of f i n i t e extent over which the gas properties change continuously, but ra p i d l y . The width of the zone turns out to be of the order of magnitude of a few i n t e r p a r t i c l e distances r e l a t i v e to the gas i n state 1. Throughout this zone, the effects of heat conduction and v i s c o s i t y cannot be ignored because of the large v e l o c i t y and temperature gradients. A discussion of these phenomena i s beyond the scope of this thesis and, furthermore, 13 i s i r r e l e v a n t since we are concerned only with the f i n a l state 2 of the gas. The discussion therefore applies only to gas out-side the t r a n s i t i o n zone. 2.3 Shock Waves in Real Monatomic Gases The theory outlined above applies to perfect gases. However, the r e s u l t s predict well the behaviour of the i n e r t monatomic gases up"to temperatures of about 8000°K. Beyond thi s point, i o n i z a t i o n and, to a lesser extent, e x c i t a t i o n must be considered i n determining the properties of state 2. Equations (2.1) to (2 .'3) s t i l l apply. However, the expressions for the i n t e r n a l energy and the equation of state must be modified. We are concerned with temperatures at which only a single stage of i o n i z a t i o n need be considered. Consider a volume V of gas which has been heated by a shock wave to such a temperature. We denote the numbers of neutral atoms, ions, and electrons by N^, N +, and N g respectively. Electrons and ions must be produced i n pairs so that N e = N + (2.17) The t o t a l number of heavy p a r t i c l e s , N, i s defined by N = N A + N + (2.18) We now introduce the degree of i o n i z a t i o n , a, N e N e a = W = Nfl I N. ( 2 ' 1 9 ) With these d e f i n i t i o n s , the equation of state for the ionized n e. . (7,9,10,11) gas i s K * * 14 pV = (N A + N + + N e)kT = (1 + a)NkT (2.20) It has been assumed that the component gases of electrons, atoms and ions are i n mutual equilibrium at temperature T. The mass density, p, i s given by PV = NAmA + N+m+ + N e me ri Nm. (2.21) A since m << m. nr. Thus e A + p .= (1 + a) pRT (2.22) since R = k/m^. The in t e r n a l energy, U, per unit volume, must now include the i o n i z a t i o n a l energy. I f the i o n i z a t i o n p o t e n t i a l of the atom i s I, U= (I NAI<T+ f N+kT + | NekT + N + l ) V"' or Here the contributions from electronic e x c i t a t i o n of both neutral atoms and ions has been neglected. The error introduced i n this way i s less than 2% for temperatures less than 16,000°K W . Equations (2.1), (2.2), (2.3), (2.22) and (2.23) now specify the state of the shock heated gas i n terms of a. The relati o n s h i p between a and the temperature and density for an ionized gas i n thermal equilibrium is provided by the Law of Mass Action or, as i t i s frequently c a l l e d when expressed in the following form, the Saha equation: Net _ Z eZ+ € - I / k T (2.24) / - a - zA Here, Z^, Z + and Z g are the p a r t i t i o n functions of the respective species, and are found for argon to be Z = 2 K e e Z. = K. A A f ..... Z + = (4 + 2 e - 2 0 6 ° / T ) K + where ' K . = ^ ^ k T J 4 v and V = volume occupied by the gas h = Planck's constant k '= Boltzmann's constant nu = p a r t i c l e mass of species i ( i = A,+ ,e) The contribution to Z^ from e l e c t r o n i c e x c i t a t i o n has been neglected, with an error of less than 11 f o r temperatures up to 15,000 oK^* **2^ . The expression for Z + includes contributions from the low-lying f i r s t excited state of the argon ion and i s v a l i d up to 2 0 , 0 0 0 o K ( 1 0 ' 1 1 , 1 2 : ) . Combining (2.20) and (2.24) * , ^ . M e - I / k T j ' / 2 C2.2S) When i o n i z a t i o n i s important, then, in addition to the Rankine-Hugoniot equations, (2.1) to (2.3), we must use the modified equation of state, (2.22), and i n t e r n a l energy, (2.23), together with Saha's equation for a, (2.24). We s h a l l c a l l t h i s complete set of equations the Augmented Rankine-Hugoniot equations. Knowing state 1 and the shock speed, they uniquely specify state 2, 2.4 Dependence of Shock Strength on Diaphragm Pressure Ratio The degree to which the test gas is shock heated depends on the shock strength which, i n a conventional shock tube, 16 depends on the diaphragm pressure r a t i o p^/p^. It i s therefore desirable to know the r e l a t i o n s h i p between P^/p^ and. the shock Mach number, M, or the shock pressure r a t i o , p2/p^. To determine t h i s dependence we must consider the processes occurring i n the driver gas when the diaphragm breaks. We s h a l l assume that the diaphragm disappears instantaneously and therefore introduces no perturbations into the flow. Driver gas then starts to move into the test section. Due to the depletion of driver gas, the pressure i n the driver section near the diaphragm drops. This decrease i n pressure propagates backwards through the driver gas as a r a r e f a c t i o n wave, or fan. The head of the r a r e f a c t i o n wave propagates into the driver gas at the sound speed, a^. After some time, t ^ , the s i t u a t i o n i s as shown i n Figure 2-2. Quantities i n the expanded driver gas behind the contact surface we denote by the subscript 3. Since there i s no momentum flux across the interface (assumed planar) between test and driver gases, the pressure across the contact surface i s continuous: p 2 = p 3 (2.26) To f i n d the dependence of shock strength on p^, we then require the r e l a t i o n between p^ and p^. For a perfect driver gas of constant r a t i o of s p e c i f i c heats, the t r a n s i t i o n from state 4 to state 3 through the r a r e f a c t i o n wave i s an isentropic process. In this case i t can be shown that • v + = c o n s t a n t (2.27) \-\ through the rarefaction w a v e ^ , where v i s the flow speed and 17 a the sound speed. Since i n i t i a l l y -V 4 = ° » (2.28) and since v 2 ~ V 3 (2.29) for the same reason that p 2 = Pg, we have that z Of I For an isentropic process f>j> = c o n s t a n t (2.31) so that, using (2 .13) , 2&V •fey _ / a* \ v* (2.32) (2.33) We then obtain the r e l a t i o n between p^ and p 2 Zh A - / \ The desired dependence of shock strength on P^/p^ i s obtained by solving (2.33) together with the Augmented Rankine-Hugoniot equations of section 2.2. For a perfect gas we fi n d the i m p l i c i t r e l a t i o n s h i p nil (2-34) This expression shows that there is a l i m i t to the value of M which can be produced for given driver and test gases. For a perfect gas, as V^/v± approaches i n f i n i t y , 18 4 a4 ^ ~ a3"V3<3— i 3 2 > V2 1 — > V s R a r e f a c t i o n Diaphragm Contact Shock Wave P o s i t i o n Surface Front Head T a i l , . (a) Wave System i n Shock Tube 1 i p 1 1 P3 .1 I *>2 • I 1 1 1 1 1 1 P i 1 1 1 1 s 1 1 (b) Pressure D i s t r i b u t i o n i n Shock Tube 0 x (c) x - t Diagram Figure 2-2 Gas Flow i n Shock Tube A f t e r Diaphragm Bursts or, to a good approximation, r i - t W ^ ' f ; C 2 . 3 5 ) To obtain strong shocks one thus requires a d r i v e r gas of high sound speed, a^, at high pressures. 2 . 5 Solution of the Equations The solution of the equations i s tedious but straight-forward for a perfect gas. For a monatomic gas including ion-i z a t i o n e f f e c ts the solution becomes much more d i f f i c u l t . We have used a d i g i t a l computer to solve the equations numerically-using the i t e r a t i o n technique described by Gaydon and H u r l e ^ . The calculated dependences of P2/p^, P2/Pi> ^2^1' anc* a o n M ' and the dependence of M on p^/p^, are shown on the following pag for argon test gas and for a driver gas of constant %^ - 1.67. These curves enable the properties of the gas i n state 2 to be predicted from the i n i t i a l state 1 and the shock Mach number. The numbers beside each curve specify the i n i t i a l gas pressure, p^, i n Torr. The lowest curve i n Figure 2-7 (a^/a^ = 3.3) pertains to shocks i n room-temperature argon driven by room-temperature helium. In this case i t i s impossible to achieve shock speeds in excess of about Mach 12.6. By heating the driver gas, however, i t s sound speed can be raised and f a s t e r shocks can be obtained as shown by the other curves of Figure 2-7. 20 1000 1 5 10 15 20 25 Mach Number Figure 2-3 Pressure R a t i o across Shocks i n Argon 21 14 J Mach Number Figure 2-4 Density and V e l o c i t y R a t i o s Across Shocks i n Argon 2 2 2 0 H 1 8 1 6 1 4 1 2 1 0 co o t-i CM 8 H 4 H T R = 2 9 3 " K = 1 0 0 Torr -j 1—i—t-—r—i—s—r 1 5 I I > I > I t < t I i I 1 0 1 5 2 0 Mach Number Figure 2 - 5 Temperature Behind Shock i n Argon Shock Mach Number Figure 2-6 Degree of I o n i z a t i o n (a) i n Shock-Heated Argon l o g 1 0 ( p 4 / p 1 ) Fi g u r e 2-7 Shock Mach Number (M) as a Function of Diaphragm R a t i o (P4/Pi) f o r Shocks i n Argon Driven by Helium 2 . 6 The Shock Heated Gas Our assumption that the gas of region 2 i s in a state of thermodynamic equilibrium deserves further comment. A perfect gas possesses only t r a n s l a t i o n a l degrees of freedom. These are excited i n the t r a n s i t i o n zone we have i d e n t i f i e d with the shock front i n section 2.1. Since energy is exchanged very e f f i c i e n t l y between p a r t i c l e s of the same mass, a state of t r a n s l a t i o n a l equilibrium i s established within a very few c o l l i s i o n s . We can now define more p r e c i s e l y the shock t r a n s i t i o n as the distance over which t h i s t r a n s l a t i o n a l equilibrium i s achieved. Real gases, however, possess i n t e r n a l as well as trans-l a t i o n a l degrees of freedom. In the t r a n s i t i o n zone a state of t r a n s l a t i o n a l equilibrium w i l l s t i l l be achieved very q u i c k l y 1 0 I f the temperature characterizing this state i s s u f f i c i e n t l y low, no other degrees of freedom w i l l be excited to a s i g n i f i c a n t extent and the state of the gas w i l l be one of thermodynamic equilibrium. However, i f this temperature exceeds about 7000°K i n argon, for example, then for a state of thermodynamic e q u i l -ibrium to be set up, some of the t r a n s l a t i o n a l energy must be d i s t r i b u t e d among the other available degrees of freedom. The c h a r a c t e r i s t i c time for this process i s c a l l e d the relaxation time and the distance over which i t occurs, the relaxation zone. A relaxation time may be defined for each such process. For diatomic gases, for example, we may speak of relaxation times for r o t a t i o n , v i b r a t i o n , d i s s o c i a t i o n , e l e c t r o n i c e x c i t a t i o n , and i o n i z a t i o n . A true temperature for the gas cannot be defined u n t i l a l l the appropriate relaxation processes have been completed' and a state of complete thermodynamic equilibrium thus established. For argon we need consider only i o n i z a t i o n a l relaxation, since the e x c i t a t i o n a l relaxation of argon atoms and ions i s (9 10 12") n e g l i g i b l e for temperatures of intere s t here v ' ' J . By the time the gas has passed through the shock t r a n s i t i o n a trans-l a t i o n a l equilibrium e x i s t s . Through' the r e l a t i v e l y i n e f f i c i e n t f13 - 191 process of atom-atom c o l l i s i o n s ^ , a few electron-ion pairs are then produced. When a s u f f i c i e n t number of electrons are a v a i l a b l e , equilibrium i o n i z a t i o n i s quickly brought about through electron-atom c o l l i s i o n s . The extent of the relaxation zone depends on the cross-sections for these processes. Some distance behind the shock front, then, a state of thermodynamic equilibrium i s reached for which the Augmented Rankine-Hugoniot equations apply. Provided the width of the relaxation zone is small compared to the shock front-contact surface separation, a useable slug of equilibrium gas s t i l l e x i s t s . In this respect, the presence of even small quantities of e a s i l y ionizable im-p u r i t i e s i s b e n e f i c i a l . These impurities can r e a d i l y supply electrons which then ionize the argon atoms, so that equilibrium i s brought about more quickly and the relaxation zone is shortened. A snapshot of a strong shock i n argon might appear as depicted i n Figure 2-8. A sharp luminous l i n e at the shock front i s followed by a dark relaxation zone and then a region of general luminosity. The sharp luminous l i n e has been found to be due to vi b r a t i o n a l spectra of such e a s i l y dissociated molecules as C ? 27 P 2' S>2 ' T 2 ^ P l ' Uniform Luminosity R e l a x a t i o n Zone (Dark) Shock Front ( F a i n t l y Luminous) Figure 2-8 Luminosity S t r u c t u r e i n Strong Shock 28 and C N.^^ these species radiate before they are destroyed by the high temperatures. In the dark region the dissociated atoms are ionized and the electrons produced acquire s u f f i c i e n t energy through c o l l i s i o n s to ionize the argon atoms. The region of f a i r l y uniform luminosity sets in when the gas has reached a state of equilibrium, and emits a spectrum c h a r a c t e r i s t i c of th i s state. In extremely pure gases, the shock front luminosity may be absent and the equilibrium r a d i a t i o n w i l l be considerably delayed. ~ ' . 2.7 Boundary Layers In deriving the shock relations in sections 2.1 and 2.2, the e f f e c t of the vessel containing the gas was ignored. Due to v i s c o s i t y , gas at the shock tube wall must have zero v e l o c i t y i n the laboratory frame of reference. S i m i l a r l y , the temperature of this gas must be that of the wa l l . Consequently, r a d i a l v e l o c i t y and temperature gradients are established i n the shock heated gas. The region near the wall where these gradients are s i g n i f i c a n t i s c a l l e d the boundary layer. The boundary layer thickness i s formally defined as the distance from the wall at which the relevant quantity achieves 99% of i t s value far from the w a l l . Shock tube boundary layers are usually f a i r l y thin com-pared to the tube radius for pressures greater than a few Torr. However, t h e i r e f f e c t can be s i g n i f i c a n t . Some of the shock-heated gas which i d e a l l y would be c o l l e c t e d between shock front and contact surface remains i n the boundary layer and i s passed 29 over by the contact surface. Due to the thermal boundary layer, the gas density in steady state must be highest near the wall since the r a d i a l pressure gradient i s small. Therefore the boundary layer can contain a large portion of the shock heated gas. This removal of gas reduces the thickness of the shock-heated gas slug between the shock front and contact surface. The boundary layer also introduces a small r a d i a l component of v e l o c i t y into the shock-heated gas. The boundary layer also has the e f f e c t of increasing the speed of the contact surface above the predicted value and attenuating the shock front. As shock-heated gas i s removed, dr i v e r gas, and hence the contact surface, must advance towards the shock front. Therefore the speed of the contact surface increases above the i d e a l value. As the driver gas expands in this way, i t s pressure, p^, drops. The shock-heated gas pressure, p 2 , must then drop as w e l l . The r e l a t i o n between p 2 and M thus requires that the shock speed decrease (see Figure 2-4). The e f f e c t of boundary layers on the properties of the retained shock-heated gas are i n general small. Due to the shock attenuation, successive a x i a l samples of gas are heated to lesser degrees. For the low attenuation usually present the r e s u l t i n g a x i a l non-uniformity i n the slug i s n e g l i g i b l e . 2.8 The E f f e c t of Shock Tube Extremities Up to this point we have assumed that the test and driver sections are of i n f i n i t e length. We now consider the r e f l e c t i o n of the shock and r a r e f a c t i o n waves from r i g i d walls, for example, 30 . the end walls of the driver and test sections. On r e f l e c t i o n from a r i g i d w a l l , the wave nature i s not changed; that i s , a shock i s r e f l e c t e d as a shock and a rarefac t i o n as a ra r e f a c t i o n . When the ra r e f a c t i o n reverses i t s d i r e c t i o n of motion on r e f l e c t i o n from the driver end w a l l , i t proceeds into gas which has been set i n motion by the incident r a r e f a c t i o n . The r e f l e c t e d wave then propagates down the shock tube with a speed given by the sum of the l o c a l flow v e l o c i t y and the l o c a l speed of sound. It must therefore eventually overtake the contact surface and the shock front i f the test section is s u f f i c i e n t l y long. When i t does so, the pressure, p 2 , pr o p e l l i n g the shock wave decreases and the shock wave attenuates. This point i s taken up again i n Chapters 5 and 6. When a shock i s r e f l e c t e d from a r i g i d w a l l , i t returns as a shock and therefore heats even more the gas which was processed by the incident shock. As a r e s u l t , the doubly-shocked gas may become highly luminous even though the incident shock front i s too weak to be luminous. This fact suggests a technique for i d e n t i f y i n g non-luminous shock fronts by r e f l e c t i n g them from an obstruction. The state of the doubly-shocked gas may be c a l -culated from the state 2 using the theory of section 2.2. As the r e f l e c t e d shock continues to move back along the shock tube, i t meets the contact surface where, i n general, i t i s refracted. The change in speed of the r e f l e c t e d shock wave on passing through the contact surface provides a convenient means of detecting the contact surface. This r e f l e c t e d shock technique was applied to obtain some results presented i n Chapter 6. CHAPTER 3 31 . APPARATUS 3.1 I n t r o d u c t i o n The c o n s t r u c t i o n o f the Smy shock tube, shown i n F i g u r e 3-1, i s d i s c u s s e d i n t h i s chapter. A d e s c r i p t i o n of i t s o p e r a t i o n has been g i v e n i n s e c t i o n 1.2. The d r i v e r s e c t i o n i s d e s c r i b e d i n s e c t i o n 3.2 and d e t a i l s are g i v e n f o r i t s f a b r i -c a t i o n . The t e s t s e c t i o n ( s e c t i o n 3.5) has an i n s i d e diameter of 2 inches and c o n s i s t s o f s e v e r a l p i e c e s which may be assembled as d e s i r e d . The b a s i c i n s t r u m e n t a t i o n used i n the i n v e s t i g a t i o n i s d i s c u s s e d i n s e c t i o n s 3.6 and 3.7. 3.2 The D r i v e r S e c t i o n The d r i v e r s e c t i o n of the Smy shock tube i s laminated from s i x - i n c h square s l a b s of l u c i t e as shown i n F i g u r e 3-2. The laminae s u p p o r t i n g the e l e c t r o d e s and gap a d j u s t i n g rod (2, 3, and 4) are bonded together w i t h a c e t i c a c i d . The lower e l e c t r o d e and c u r r e n t l e a d are glued i n t o a r e c e s s m i l l e d i n t h i s composite l u c i t e s e c t i o n . The upper e l e c t r o d e c u r r e n t l e a d i s g l u e d i n t o a recess i n the l u c i t e back p l a t e ( 1 ) . O-rings s e a l the remaining j o i n t s . The whole assembly i s b o l t e d between brass p l a t e s (I and II) and can be taken apart to r e p l a c e the T e f l o n i n s u l a t i n g sheet which d e t e r i o r a t e s a f t e r s e v e r a l hundred d i s c h a r g e s . A s m a l l h o l e through the d r i v e r s i d e w a l l (see S e c t i o n A-A') allows e v a c u a t i o n and f i l l i n g o f the d r i v e r . F i g u r e 3-4 shows a photograph of the assembled d r i v e r . In the p r e s e n t work, the d r i v e r s e c t i o n was f i l l e d w i t h helium at D r i v e r S ection Test S e c t i o n Dump Chamber Backstrap Gas I n l e t /Flanged Copper Pipe if /Lk. Diaphragm ; 0 - r i n g Glass Pipe C a p a c i t o r Bank (50 jifarad) R inging P e r i o d — '21 p.sec 4 -1 Logarithmic Decrement * 1.7*10 sec Pressure Gauges Vacuum Pump Figure 3-1 Schematic Diagram of V e r t i c a l Section through Shock Tube A x i s F i g u r e 3-2 D r i v e r S e c t i o n of Shock Tube (Scale: Two-Thirds F u l l S i z e ) 34 atmospheric pressure. A i r was used i n Smy's. o r i g i n a l design but the use of helium resulted i n faster shocks. Since helium has a low breakdotm voltage, a triggerable a i r spark gap was required i n series with the d r i v e r spark gap, which, in our experiments, had a separation of 2 cm. 3.3 The Discharge C i r c u i t The discharge c i r c u i t i s shown schematically i n Figure 3-3. The capacitor bank, C, consists of f i v e NRG Type 203 10-microfarad capacitors i n p a r a l l e l , and i s charged by a Sorenson Model 1020-30 power supply, P, to a voltage of 10 ± 0.1 k i l o v o l t s unless otherwise stated. A trigger spark activated by the pulse generator, T, and transformer, X, produces a small degree of i o n i z a t i o n i n the a i r spark gap, A, causing i t to conduct. The driver spark gap, D, then breaks down, com-pl e t i n g the c i r c u i t . T represents a thyratron pulse generator and voltage doubling c i r c u i t which have been described (22 231 elsewhere v *• J . It may be triggered manually or by a p o s i t i v e pulse of amplitude greater than 20 v o l t s . 3.4 The Diaphragm The diaphragm i s securely clamped between the driver and a flanged two-inch copper pipe which forms the f i r s t section of the shock tube (Figure 3-1). 0-rings on each side of the diaphragm grip i t and provide a vacuum-tight s e a l . After discharge the clamp is lossened i n order to replace the diaphragm and then retightened. D A: A i r Spark Switch C: C a p a c i t o r (.50 jifarad) D: Shock. Tube D r i v e r S e c t i o n P: 20 K i l o v o l t Power Supply T: T r i g g e r Pulse Generator X: I s o l a t i o n and Step-Up Transformer L: Current Leads (0.063 inc h t h i c k copper sheet 4 inches wide and 3 f e e t long) F i g u r e 3-3 D e t a i l s of Discharge C i r c u i t 36 In Smy's o r i g i n a l shock tube, mylar diaphragms about 0.0015 inch thick were used. Upon f i r i n g the discharge, the diaphragm burst and tore into small pieces which flew down the tube. These pieces burned i n f l i g h t and l e f t a dense black deposit on the shock tube w a l l . In our experiments the burning b i t s of mylar also damaged delicate pressure probes which were used as measuring devices. A h a l f - i n c h thick glass end plate used i n some experiments was severely chipped by the mylar fragments aft e r only three or four shots. To avoid this kind of damage and to improve the cleanliness of the shock tube, the mylar diaphragms were replaced by thin sheets of shim brass. These remained i n one piece a f t e r rupturing and l e f t only a s l i g h t deposit which could e a s i l y be removed with t i s s u e . The mylar deposit could not be completely removed without using acid. The brass diaphragms consisted of three-inch squares of automotive shim stock scribed with a sharp knife along the central two inches of each diagonal to a depth s l i g h t l y less than the diaphragm thickness. The depth of the scores - varied by varying the force exerted on the knife - had no noticeable e f f e c t on the shock speed provided the grooves were deep enough that the diaphragms opened completely along both diagonal cuts. A photograph of diaphragms before and a f t e r bursting i s shown i n Figure 3-5. The four triangular shaped sections of the burst diaphragm are c a l l e d "petals". I n i t i a l l y the diaphragms were clamped against the open end of the flanged copper pipe but for diaphragms thinner than 0.008 inch, the petals tore o f f . To prevent t h i s , the pipe opening was f i t t e d with a brass i n s e r t Figure 3-4 Two Views of The D r i v e r Section Figure 3-5 Diaphragms Before and A f t e r B u r s t i n g 38 which had a 1.4 i n c h square opening at the diaphragm and tapered t o merge w i t h the c i r c u l a r p i p e w a l l s at a d i s t a n c e two inches downstream. The diaphragm p e t a l s then f o l d e d back along the square edges and remained att a c h e d except f o r diaphragms t h i n n e r than 0.003 i n c h and d i s c h a r g e v o l t a g e s i n excess of 13 k i l o v o l t s . The diaphragms were centered on the shock tube a x i s by f i t t i n g them i n t o a t h r e e - i n c h square l o c a t i n g aperture a t t a c h e d t o the f l a n g e d copper p i p e . Diaphragms ranging i n t h i c k n e s s from 0.002 i n c h to 0.015 were t r i e d . A l l opened s a t i s f a c t o r i l y when p r o p e r l y s c r i b e d . Unless otherwise s t a t e d , 0.005 i n c h t h i c k diaphragms were used. 3.5 The T e s t S e c t i o n In d i s c u s s i n g the t e s t s e c t i o n , a l l a x i a l d i s t a n c e s are measured from the diaphragm. T h i s s e c t i o n c o n s i s t s of a tube about two meters long of two-inch i n s i d e diameter through which the shock wave propagates. The tube i s c o n s t r u c t e d from lengths of copper p i p e , pyrex p i p e , and machined l u c i t e s e c t i o n s , clamp-ed at the j o i n t s and s e a l e d w i t h 0 - r i n g s . The f i r s t p a r t of the t e s t s e c t i o n i s a f l a n g e d copper pipe mentioned p r e v i o u s l y . Argon t e s t gas i s admitted through a s m a l l hole i n the s i d e of t h i s p i p e v i a a needle v a l v e . Seven r a d i a l holes spaced at two . i n c h i n t e r v a l s were bored through the w a l l to admit p r e s s u r e probes. The f i r s t h o l e i s f i v e inches from the diaphragm. The pipe l e n g t h spanned by these holes i s e n c l o s e d i n a c a s i n g made of l u c i t e s l a b s bonded together and to the copper p i p e . The p i p e was e l e c t r i c a l l y grounded i n order to s h o r t - c i r c u i t any 39 currents which might pass from the dr:.ver electrodes through the intervening conducting gas. A s p e c i a l l y constructed portion of the shock tube i s shown i n Figure 3-6. Portholes, which could be plugged when not i n use, allowed access to the shock tube i n t e r i o r . The f i r s t port was usually occupied by a pressure probe. On the opposite side of this section, a s l i d i n g mount allowed the a x i a l separation between a pressure probe and the second port to be varied continuously over a three-inch extent. Several l u c i t e discs 1.5 inch thick and 6 inches in diameter with two-inch a x i a l holes could be inserted as desired between glass sections of the shock tube. A r a d i a l hole i n each disc allowed a pressure probe to be inserted. The shock tube was terminated i n a so-called dump chamber (see Figure 3-1) constructed from a four-inch length of four-inch diameter brass pipe". A removable end plate allowed the shock tube to be cleaned with a long brush. Separate connections to the vacuum pump and pressure gauges were made through the side-wall of the dump chamber. The shock tube was evacuated with a Cenco Hyvac 14 mechanical pump. The lowest pressure achieved was 0.5 m i l l i t o r r and the leak rate was about 20 m i l l i t o r r per hour. Pressures were measured with two Edwards Vacustat gauges covering the ranges 0 to 1 Torr and 0 to 10 Torr and with two Edwards capsule d i a l gauges covering the ranges 0 to 40 Torr and 0 to 760 Torr. After evacuation the tube was flushed with argon before f i l l i n g to the desired pressure. / N e o p r e n e G a s k e t s 3L P l u g L u c i t e P l u g P y r e x P o r t 1 P o r t 2 P i p e D i a m e t e r L u c i t e T 4.25 i n c h s q u a r e S l o t 0.25" w i d e , 3" l o n g r—: i B r a s s ' ' S l i d i n g Mount f o r P r e s s u r e P r o b e O F i g u r e 3-6 H o r i z o n t a l A x i a l S e c t i o n t h r o u g h S p e c i a l T e s t S e c t i o n 41 3.6 Instrumentation The shock front was detected with p i e z o e l e c t r i c pressure probes. The design c r i t e r i a and construction of the probes are discussed i n the Appendix. Application of an a x i a l stress to the active element of the probe produces an e l e c t r i c a l s i g n a l which is recorded on an oscilloscope. A general purpose probe i s shown i n Figure 3-7. A s i m i l a r t r i g g e r probe was usually mounted i n the f i r s t port of the shock tube section of Figure 3-6. Shock speeds could be accurately measured i n the following way. The signal from the trigger probe was amplified by one channel of a Tektronix-Type 1A1 Plug-in Unit and then used to trig g e r timebase B of a Type 545A oscilloscope. The delayed trigger pulse was used to trigger timebase A which displayed the signal from a second pressure probe some distance downstream of the f i r s t . This t i m e - o f - f l i g h t measurement,, when corrected for the delays inherent in the probes (see Appendix), was used to calculate an average shock speed over the distance between the probes. A smear camera (designed and constructed by J . P. Huni) with writing speed variable between 45 ysec/cm and 4 ysec/cm was used-to follow the progress of luminous shock wave features. The image was recorded on Polaroid f i l m and presents an x-t diagram of the flow i n the shock tube. The driver discharge was triggered by a pulse generated when the smear camera rotating mirror had achieved a preselected speed and was in the required p o s i t i o n to record the event of in t e r e s t . The smear camera s l i t was imaged on the f i l m plane by a f r o n t - s i l v e r e d concave BNC Connector i Brass Case . I Soft P l a s t i c . Tube to F i g u r e 3 - 7 S i m p l i f i e d Diagram of Pressure Probe and Housing mirror which obviated the need for a lens system inside the camera. A p a i r of 15 cm f o c a l length lenses served to image the event onto the smear camera s l i t . 3.7 E l e c t r i c a l Measurements To measure the driver voltage, a Tektronix P6013A 1000:1 po t e n t i a l d i v i d e r was connected across the electrode leads d i r e c t l y below the driver section. The output from the p o t e n t i a l divider was displayed on a calibrated oscilloscope. To measure the discharge current, a Rogowski c o i l was (20 211 used v ' J . This device consists of a t o r o i d a l l y wound multi-turn c o i l whose minor radius i s small compared to i t s major radius. The rapidly-varying current to be measured i s passed through the torus opening. Applying Ampere's law to a closed path threading the turns of the c o i l and assuming the v a r i a t i o n of magnetic induction to be small over the area of an i n d i v i d u a l turn, one obtains for the output voltage of the c o i l V o u t "= - n A A . j l where n = number of turns per unit length of c o i l A = area of the i n d i v i d u a l turns I - discharge current to be measured yo - magnetic permeability constant t - time Usually the output i s integrated to give the current, I. Providing that for a l l frequencies of i n t e r e s t , , LcO << R and (OCR >> 1, where R and C are the resistance and capacitance of the integrating c i r c u i t and L i s the self-inductance of the c o i l , one then obtains 44 V f - - n T (3.2) Vout TIC The Rogowski c o i l used i n thi s work was constructed from a suitable length of RG 65A/U delay cable with the outer con-ductor removed. Before t o r o i d a l l y deforming th i s c o i l i t was wrapped with a 0.001 inch thick piece of brass shim stock, which did not close on i t s e l f e l e c t r i c a l l y to permit penetration of the magnetic f i e l d . This s h i e l d was e l e c t r i c a l l y grounded to reduce the e l e c t r o s t a t i c coupling between the c o i l and current leads. To reduce further the ef f e c t of such capacitative coupling, the c o i l signal was amplified d i f f e r e n t i a l l y . For the c o i l used, the current measured i s given by I -- 1.1$ -10* V o u t j%ff-45 CHAPTER 4 THE DRIVER 4.1 Introduction In t h i s chapter the nature of the d r i v i n g force responsible for the high shock speeds obtained i n the Smy shock tube i s investigated. A technique i n which we use the diaphragm opening process to measure the pressure of the driver gas i s then discussed. 4.2 The Driving Mechanism Smy^'^ attributed the high shock speeds to electro-magnetic forces produced by the "backstrap" configuration. There remains, however, another possible d r i v i n g mechanism. In Chapter 2 i t was shown that very fast shocks can be produced i n pressure-driven shock tubes by using a driver gas of high sound speed at high pressure. JustTthis s i t u a t i o n may be produced by the e l e c t r i c a l discharge which c l e a r l y must increase the tempera-ture, and hence the pressure and sound speed of the driver gas, by Joule-heating. To investigate the r e l a t i v e importance of the two possible shock d r i v i n g mechanisms several experiments were car r i e d out. The existence of a downstream-directed electromagnetic d r i v i n g force obviously requires the presence of the "backstrap" in the absence of any other source of magnetic f i e l d at the s i t e of the discharge. Hence the removal of the "backstrap" should remove the d r i v i n g force and so have a very noticeable e f f e c t on the shock speed. The "backstrap" was eliminated by conducting 46 the current from the upper dr i v e r electrode through a copper s t r i p down each side of the discharge chamber. In this config-uration there i s no downstream-directed electromagnetic force and so the shocks must be driven by the k i n e t i c pressure of the d r i v e r gas heated by the discharge. Measurements of shock speed with and without the "backstrap" are compared i n Figure 4-1 for an argon pressure of 1 Torr at x = 126 cm. The removal of the "backstrap" i s seen to have only a small e f f e c t on the shock speed. Another experiment was conducted to confirm that e l e c t r o -magnetic forces are n e g l i g i b l e . Since an electromagnetic d r i v i n g force would depend on the square of the discharge current, the speed of shocks driven i n this way should depend strongly on the current. With the "backstrap" i n place, the discharge current was reduced by ins e r t i n g series inductors into the discharge c i r c u i t . Each inductor consisted of a c i r c u l a r c o i l of two-inch wide copper s t r i p supported by a six-inch diameter plywood d i s c . Inductors of one and two turns reduced the peak discharge current by 21% and 49% respectively. To check that the f i e l d configuration around the inductors did not cause erroneous current measurements, the Rogowski c o i l was moved farther away from the inductive loop and i t s orientation r e l a t i v e to the plane of the loop was altered. No change in the current waveform resulted. The results of these measurements, shown in Figure 4-2, demonstrate that the shock speed does not depend on the discharge current. It may be concluded, then, that electromagnetic forces 5 H o CO > 4 s QJ <D ft co ,M o o CO T e s t Gas P r e s s u r e = 1 T o r r E r r o r B a r s R e p r e s e n t S t a n d a r d D e v i a t i o n s f o r a t l e a s t 3 S h o t s O B a c k s t r a p x S i d e s t r a p s o 4 —T— 11 — S — 12 —r— 13 - i — 14 — r 15 7 8 9 -I— ID C a p a c i t o r B a n k V o l t a g e ( K i l o v o l t s ) F i g u r e 4-1 S h o c k S p e e d s W i t h a n d W i t h o u t t h e B a c k s t r a p 49 p l a y a n u n i m p o r t a n t r o l e a n d t h a t t h e h e a t i n g o f t h e d r i v e r g as s p e e d s . I n a l l s u b s e q u e n t m e a s u r e m e n t s , t h e " b a c k s t r a p " c o n f i g -u r a t i o n was u s e d . ; 4.3 D e t e r m i n a t i o n o f t h e D r i v e r Gas P r e s s u r e H a v i n g e s t a b l i s h e d t h a t t h e s h o c k s a r e d r i v e n b y gas r a i s e d t o h i g h t e m p e r a t u r e b y t h e e l e c t r i c a l d i s c h a r g e , i t i s d e s i r a b l e t o d e t e r m i n e t h e m a g n i t u d e o f t h e d r i v i n g p r e s s u r e . The o b s e r v a t i o n o f t h e d i a p h r a g m o p e n i n g p r o c e s s p r o v i d e s a c o n v e n i e n t means o f m e a s u r i n g t h i s q u a n t i t y . b o t h d i a g o n a l c u t s and t o o p e n u n d e r t h e a c t i o n o f a c o n s t a n t i s o t r o p i c p r e s s u r e , p . The p e t a l s a r e assumed t o b e h a v e l i k e p l a t e s f r e e l y h i n g e d a t t h e e d g e s o f t h e s q u a r e - m o u t h s h o c k t u b e i n s e r t . The w o r k done i n s p l i t t i n g t h e d i a p h r a g m a nd b e n d i n g t h e p e t a l s i s n e g l e c t e d . The e q u a t i o n o f m o t i o n o f a p e t a l i s t h e n b y t h e e l e c t r i c a l d i s c h a r g e i s r e s p o n s i b l e f o r t h e h i g h s h o c k The d i a p h r a g m i s assumed t o s p l i t i n s t a n t a n e o u s l y a l o n g I (4.1) w h e r e P I i s t h e a p p l i e d t o r q u e i s t h e moment o f i n e r t i a o f t h e p e t a l a b o u t t h e a p p r o p r i a t e a x i s i s t h e a n g l e t h r o u g h w h i c h t h e p e t a l moves i s t i m e ( t = 0 a t e = 0) 6 t Figure 4-3 Diaphragm Petal Geometry Since the t o t a l force acting on a petal i s p.area = ph , 3 the torque about the axis 00' i s ph /3 (Figure 4-3), while the 7 moment of i n e r t i a of the petal about this axis i s mh /6, where 2 m i s the petal mass given by m = ph d for a diaphragm of mass density p and thickness d. Equation (4.1) then becomes d a 0 - ZP d t 1 " fhd Integrating t h i s equation subject to the conditions 6 = 0 , de/dt = 0, at t = 0 y i e l d s e = Now the time, t , for a diaphragm to open completely i s obtained by setting e = IT/2: L o P Z \ p J (4.2) (4.3) (4.4) (4.5) A measurement of t then provides a value f o r p. op r A The opening time, t Q p , was measured with the smear camera set up of Figure 4-4. The camera s l i t was focused on a l i n e D r i v e r Diaphragm Shock Tube (a) Experimental Arrangement Glass P l a t e F r o n t - S i l v e r e d M i r r o r Lens / Entrance S l i t of Smear Camera Region of Diaphragm Observed with Smear Camera (b) Diaphragm Figure 4-4 Arrangement f o r Observing Diaphragm Opening Figure 4-5 Smear Photograph of Diaphragm Opening passing through the center of the diaphragm at 45° to each of the diagonals. Light from the hot driver gas illuminated the opening diaphragm. A t y p i c a l smear photo i s shown in Figure 4-5. Figure 4-6 shows the dependence of t on the capacitor bank voltage, V , for diaphragms 0.005 inch thick. One would expect • W "v. the driver pressure to be proportional to the capacitor bank 2 energy dissipated i n the driver gas ( i . e . p V ), except at low bank energies (V - 10 v o l t s ) , where the thermal energy of the cold driver gas i s comparable to the e l e c t r i c a l energy added, and at high energies, where conductive and radiative energy losses become appreciable. The p r o p o r t i o n a l i t y of t to V c^ i s thus i n agreement with equation (4.5). The p r o p o r t i o n a l i t y 1/2 between t and d ' , for constant bank voltage shown in Figure 4-7 also agrees with the p r e d i c t i o n of equation (4.5). Figures 4-6 and 4-7 y i e l d the empirical re s u l t t = c d 1 / 2 / V (4.6) „- op c -1/2 where c = 9.28 volt-sec-cm ' . Moreover, since Figures 4-6 and 4-7 seem to v e r i f y the adequacy of the theory, a single measure-ment of t for a diaphragm of known thickness serves to determine p. The quantity, p, however, may not be the true pressure in the d r i v e r , p^. Because of complex flow patterns set up in the driver as the diaphragm opens, the pressure e f f e c t i v e i n opening the p e t a l s , p, may i n fact be considerably less than the s t a t i c driver pressure, p^, immediately before the diaphragm bursts. Furthermore, since the e x i t v e l o c i t y of driver gas 53 Y 0.05 0.10 0.15 V" 1 ( ( K i l o v o l t s ) - 1 ) F igure 4-6 Dependence of Diaphragm Opening Time on Voltage depends on the driver sound speed (see Chapter 2, Equation (2.30)), one might expect the relationship between p and p^ to depend on t h i s quantity also. To determine the r e l a t i o n between p and p^, the experi-ment shown i n Figure 4-8 was set up. The driver was evacuated and then pressurized slowly u n t i l the diaphragm burst at pressure p^. At that time, laser l i g h t passed through the center of the diaphragm, activating a photodiode which triggered a Tektronix 551 dual beam oscilloscope. Light from an intense incandescent source passed through the diaphragm opening and was monitored by a photomultiplier at the end of the shock tube. The photomultiplier signal reached a maximum value when the diaphragm was f u l l y open. A pressure transducer ( A t l a n t i c Research Corporation Model LD-15) mounted externally against the shock tube wall produced a signal when the wall was struck by the diaphragm petals - that i s , when the diaphragm was f u l l y open. A simultaneous record of the photomultiplier and transducer signals i s shown i n Figure 4-9. The two methods of observing the diaphragm opening time are i n agreement. Results of many measurements (Figure 4-10) show that -1/2 -1/2 t d ' = Ap. ' with A determined by the method of least op ^4 ' 1 / 2 - 1 squares to be 5.36 g ' cm . Using equation (4.5), one then finds that p 4 = 2.55p (4.7) That i s , the actual d r i v e r pressure i s 2.55 times larger than the pressure e f f e c t i v e i n opening the diaphragm. Photodiode 5 6 Pressure L i g h t Pipe D r i v e r Gauge Gas I ,Mirror To Scope Movie T r i g g e r Sun-Gun Scope Upper Beam 1 He-Ne Lase: Pressure Transducer Shock Tube Evacuated to about l T o r r Dump Chamber Glass End-P l a t e -f F r o n t - S i l v e r e d M i r r o r <4> Le ns Scope «s-Lower-Beam RCA 93IA P h o t o m u l t i p l i e r F i g u r e 4-8 Arrangement f o r Observing Diaphragm Opening wit h Cold P r e s s u r i z e d D r i v e r Pressure Transducer 5 V/div Photomultiplier 20 mV/div r — » t Diaphragm Diaphragm F u l l y Begins to Open Break Time 50 usec/div 140 Figure 4-9 Photomultiplier and Pressure Transducer Records of Diaphragm Opening © Argon Driver Gas X Helium Driver Gas 120 ftlOO • 80-. 10 —i— .12 14 P^*** ((pounds per square inch) °' 5) Figure 4-10 Diaphragm Dynamics for Cold Driver Gas 58 The results of Figure 4-10 show that the diaphragm opening time i s independent of the cold driver sound speed for a three-fold change i n this quantity since argon and helium gave the same opening times. The sound speed of the discharge-heated driver was also altered by d i l u t i n g the helium driver gas with 12% a i r by volume. (At the gap setting used, the discharge would not " f i r e " f or a greater amount of a i r ) . Increasing the mass density of the driver i n this way decreases the sound speed by roughly 25%. No s i g n i f i c a n t change i n the diaphragm opening time occurred. From these two pieces of evidence i t appears that the diaphragm opening time does not depend strongly on the driver sound speed and so may be used as a measure of the driver pressure. Using (4.5), and (4.6) and (4.7), i t i s now possible, from the capacitor bank voltage, to calculate the driver gas pressure: 2 p. = 0.332 V atmospheres (4.8) 4 c (V i n k i l o v o l t s ) c This empirical r e s u l t i s of course of l i t t l e s i g n i f i c a n c e unless one i s sure that the pressure i n the driver i s reasonably uniform. Two simple experiments indicated that this condition i s probably achieved. The smear camera pictures of the diaphragm opening show that both l a t e r a l petals open i n about the same time. Rotating the image of the smear-camera s l i t through 90° to observe the top and bottom petals showed that they opened i n the same time as the l a t e r a l petals. This re s u l t indicates that the pressure i s uniform over the driver cross-section. 59 Another i n d i c a t i o n of the driver uniformity was obtained as follows. The time between i n i t i a t i o n of the discharge and the instant at which the diaphragm begins to break, the diaphragm rupture time, was measured by placing a photodiode on the shock tube axis behind a glass plate sealing the dump-chamber. The f i r s t driver luminosity transmitted by the diaphragm activated the diode. The current waveform was simultaneously monitored. A t y p i c a l oscilloscope record i s shown in Figure 4-11. The v a l i d i t y of rupture times measured in t h i s way was checked by recording simultaneously the rupture and opening times with the smear camera. The basic arrangement of Figure 4-4 was used. In addition, l i g h t from the discharge was transmitted v i a a l i g h t pipe to the edge of the smear camera s l i t producing a streak on the f i l m along side the image of the diaphragm opening. The time i n t e r v a l between the i n i t i a t i o n of the discharge and the breakage of the diaphragm measured i n this way agreed with the photodiode measurement of the rupture time. The rupture time measurements show that the driver gas i s heated at constant volume since the diaphragm does not begin to break u n t i l the discharge current has f a l l e n to a small value. To obtain an i n d i c a t i o n of the driver gas uniformity, the rupture time was compared with the time at which the diaphragm f i r s t feels the e f f e c t of the discharge. A pressure probe inserted through a Lucite plate sealing the driver section showed that a shock wave generated by the discharge reaches the diaphragm very quickly. The a r r i v a l time of this shock i s compared with the rupture time i n Figure 4-12. The Time 10 p.sec/div Current Waveform 5 V/div Photodiode S i g n a l Discharge Voltage = 10 KV Diaphragm Thickness = 0.005 inch Diaphragm Breaks Discharge Begins Figure 4-11 Current Waveform and Diaphragm Rupture Time 5 Time f o r D r i v e r Shock to Reach Diaphragm A [ Diaphragm Rupture Time O I I 0 0.05 , 0.10 , 0.15 0.20 V~ ((KV) ) Figure 4-12 Times C h a r a c t e r i s t i c of Discharge and Diaphragm 61 " a r r i v a l " times are mu'cli shorter than the diaphragm rupture times. This fact indicates that there i s ample time for conditions i n the driver to become f a i r l y uniform before the diaphragm breaks. It therefore seems probable that the d r i v e r uniformity approaches that i n conventional pressure driven shock tubes. 62 CHAPTER '5 COMPARISON OF SHOCK TUBE OPERATION WITH IDEAL THEORY 5.1 I n t r o d u c t i o n In the l a s t chapter we showed that the Smy shock tube i s a pr e s s u r e - d r i v e n shock tube and measured the d r i v e r pressure, p^ (equation (4. 8 ) ) . We now consider the problem of p r e d i c t i n g the shock speed from the i n i t i a l parameters, V , the c a p a c i t o r v o l t a g e , and p^, the t e s t gas pressure. The operation of the shock tube i s compared w i t h the i d e a l theory developed i n Chapter 2 f o r one-dimensional gas flow i n the f o l l o w i n g manner. The t h e o r e t i c a l dependence of shock speed on diaphragm pressure r a t i o , V/^/Vi* i s c a l c u l a t e d w i t h the sound speed r a t i o , a^/a^, as parameter, and t h e o r e t i c a l curves of shock speed versus l o g 1 0 ( p ^ / p 1 ) are p l o t t e d . Then, f o r a given d r i v e r v o l t a g e , shock speeds are measured as a f u n c t i o n of downstream pressure, p^. In order t o b r i n g the set of e x p e r i -mental p o i n t s i n t o coincidence w i t h one of the t h e o r e t i c a l curves, i t i s necessary to assume a d r i v i n g p r e s s u r e , p^. Unique values of p| and a| are determined i n t h i s way f o r s e v e r a l d i f f e r e n t bank v o l t a g e s . (a^ denotes the estimated value of the d r i v e r sound speed obtained by matching the experimental and t h e o r e t i c a l curves as described.) The r e s u l t i n g estimates of d r i v e r pressure are then compared w i t h the values determined from the diaphragm opening times (see Chapter 4 ) . However, since the shock speed i s not constant along the tube, a s l i g h t problem a r i s e s i n s e l e c t i n g the speed to be used 63 i n the above procedure. The maximum speed was chosen because i t i s unique. It was therefore necessary to locate the p o s i t i o n of the maximum shock speed for a var i e t y of downstream pressures, p^, and driver conditions. These measurements are described i n the next section, and i n the following section, the estimated values of p^ are compared with the values of p^ found i n Chapter 4 . F i n a l l y , the " e f f i c i e n c y " of the shock tube i s discussed. 5.2 Dependence of Shock Speed on A x i a l P o s i t i o n Six pressure probes were inserted along the shock tube f l u s h with the inside w a l l . Their signals were simultaneously recorded on two Tektronix oscilloscopes, a 545A and a dual beam 551, using three Type 1A1 Plug-in Units operated i n the chopped mode to give six channels. A photodiode, activated by l i g h t from the dr i v e r discharge when the diaphragm began to break, triggered one oscilloscope and a calib r a t e d delay unit which triggered the second oscilloscope at an appropriate time. The time lapse between probe signals was then used to calculate the average shock speed over the i n t e r v a l between probes. The shock was assumed to have this speed midway between the probes. The v a r i a t i o n of shock speed with a x i a l p o s i t i o n over the f i r s t 1.6 meter length of the shock tube was measured i n this way. Joining the points with a smooth curve enabled a good estimate of the maximum shock speed and i t s location to be made. The re s u l t s of these measurements are shown i n Figures 5-la and 5-lb. The features of these graphs merit some discussion. Over the i n i t i a l period of i t s f l i g h t , the shock X j i i i i i i i i 0 20 40 60 80 100 120 140 160 Distance (cm.) Figure 5-la) Dependence of Shock Speed on Distance from the Diaphragm 66 front i s seen to accelerate. After reaching a maximum speed, the p o s i t i o n of which depends on the downstream pressure, the shock wave then decelerates. The acceleration has been attributed to effects of the diaphragm opening^ 2 4 » 2 5). In the ideal theory, the diaphragm i s assumed to disappear instantaneously and the flow i s assumed to be one-dimensional. The shock wave then maintains a well-defined speed determined by the pressure r a t i o , p^/p^, and the sound speed r a t i o , a^/a^. In r e a l i t y , however, the diaphragm opening process requires a f i n i t e time, during which the driver gas i s expelled through an aperture of increasing size at a rate determined by P^/p^ and a^. Furthermore, the flow i s three-dimensional because i n i t i a l l y there are r a d i a l components in the flow v e l o c i t y . Driver gas passes through the aperture at a pressure determined by i t s rate of expulsion, and expands to f i l l the tube cross-section. As the diaphragm opening grows in s i z e , the expelled driver gas need expand less to f i l l the.tube, so i t s net drop i n pressure i s l e s s . The r a d i a l component of the flow v e l o c i t y i s also reduced. The r e s u l t i n g pressure gradient i n the expelled driver gas accelerates the shock. The deceleration of the shock i s due to several phenomena. Viscous and thermal boundary layers near the wall of the shock tube cause the shock wave to attenuate as discussed i n Section 2.7. Another e f f e c t can also influence the shock speed. When the diaphragm breaks, a rar e f a c t i o n wave travels backwards through the driver gas, i s r e f l e c t e d at the end wall of the driver and then proceeds down the shock tube i n pursuit of the 67 shock front. When the r e f l e c t e d r a r e f a c t i o n wave overtakes the contact surface, the driver gas pressure i s reduced below the value required to keep the shock heated gas moving at constant speed ( i . e . drops below P 2) • The shock speed then drops, slowly at f i r s t since the v e l o c i t y and. pressure gradients in the r e f l e c t e d r a r e f a c t i o n wave are usually small provided the overtaking does not occur too close to the dri v e r section. For constant driver conditions , the distance from the diaphragm at which this occurs i s shorter the lower the shock speed. This process probably accounts for the larger attenuation observed for shocks at test gas pressures above about 5 Torr (Figure 5-1). Below 5 Torr, the above process either did not occur or was just beginning to occur i n the length of shock tube studied. In addition, cooling of the hot driver gas as i t flows down the tube may reduce i t s pressure and sound speed and con-tribu t e to the attenuation. If the i n i t i a l shock acceleration i s due to the diaphragm opening process as described, then one would expect the distance, L, over which the shock accelerates to be of the order of the distance t r a v e l l e d by the shock front during the opening time, t : i . e . op L ~ v 't (5.1) max op v where v i s the maximum shock speed. The measurements of max r Figure 5-1, made with diaphragms 0.005 inch thick and a bank voltage of 10 KV, support this expectation and give L = 1 . 3 v m Q v t ' (5.2) max op 68 Figure 5-2 shows that an increase i n t ,. caused by increasing the diaphragm thickness while maintaining the capacitor voltage at 10 KV, r e s u l t s in an increase i n L. Moreover, the r e s u l t i n g increase i n L i s consistent with equation (5.2), as shown i n Figure 5-3 where the measurements for both 0.005 inch and 0.010 inch thick diaphragms are seen to f i t the same straight l i n e . The dependence of L on driver gas pressure, p^, was checked also. The driver pressure was reduced from 33 atmospheres, as i n the above measurements, to 16 atmospheres by decreasing the i n i t i a l helium driver gas pressure from 1 atmosphere to 100 Torr, keeping the capacitor .voltage at 10 KV. The r e s u l t s (Figure 5-4) indicate that the p o s i t i o n of maximum shock speed, L, i s independent of the driver pressure, p^. These measurements strongly suggest that the i n i t i a l acceleration of the shock wave i s attributable to the f i n i t e time required for the diaphragm to open completely, and are i n q u a l i t a t i v e agreement with the results quoted by Simpson et a l / 2 5 ' 5.3 Comparison of Shock Tube Operation with Ideal Operation It i s now possible to compare the observed maximum shock speeds with the speeds predicted by the theory discussed i n Chapter 2 (Sections 2.3 and 2.4), as outlined i n Section 5.1. We define p^ as the pressure e f f e c t i v e i n driving the shock, admitting that p| may d i f f e r from the pressure obtained from the diaphragm opening times. The meaning of p^ i s c l a r i f i e d below. Having determined the maximum shock speed as a function o — 1 1 1 1 j 1 1 I 0 4 0 80 120 160 D i s t a n c e (cm.) F i g u r e 5-2 D e p e n d e n c e o f S h o c k Speed on P o s i t i o n f o r D i f f e r e n t D i a p h r a g m T h i c k n e s s e s 100 " 0 10 2 0 - 3 0 40 50 60 70 . ^ a x ^ o p ( c m-> Figure 5-3 Dependence of L on the Product v * t © D r i v e r P r e s s u r e = 33 a t m o s p h e r e s D i a p h r a g m O p e n i n g T i m e = 100 \isec A D r i v e r P r e s s u r e = 16 a t m o s p h e r e s D i a p h r a g m O p e n i n g T i m e = 150 [ i s e c D i a p h r a g m T h i c k n e s s = 0.005 i n c h 1 1 1 1 1 1 i 1— 0 4 0 , 8 0 120 160 D i s t a n c e (cm.) F i g u r e 5-4 D e p e n d e n c e o f S h o c k S p e e d on P o s i t i o n f o r D i f f e r e n t D r i v e r C o n d i t i o n s 72 of test gas pressure, p^, for constant capacitor bank voltage, we then plo t the maximum Mach number against log^QCp^/p^) assuming i n i t i a l l y that p| i s 1 atmosphere. The r e s u l t i n g curve i s then compared with t h e o r e t i c a l curves calculated from the theory of Sections 2.3 and 2.4. A v a r i e t y of such curves f o r d i f f e r e n t assumed values of and a| (the e f f e c t i v e driver gas sound speed) are p l o t t e d . The experimental curve i s then s h i f t e d along the l o g 1 0 ( p | / p 1 ) - a x i s u n t i l i t coincides with one of the t h e o r e t i c a l curves. The values of p^ and a^ characterizing this t h e o r e t i c a l curve we then c a l l the e f f e c t i v e d r i v e r gas pressure and sound speed r e s p e c t i v e l y , because they are the values which we must substitute into the theory i n order to predict the observed shock speeds. The antilogarithm of this s h i f t gives the e f f e c t i v e driver gas pressure, , i n atmospheres. This procedure i s shown i n Figures 5-5 to 5-7 for several capacitor bank voltages and the results are co l l e c t e d i n Table I below. Bank E f f e c t i v e E f f e c t i v e Driver l o S i n ^£ Voltage Driver Sound Pressure p. Pressure Speed V c P i a« p 4 (KV) (atm) (m/sec) (atm) 7 5.3 2750 10 10.5 3500 14 15.1 4250 Table I Shock Tube Driver Conditions 16.9 0.59 33.2 0.67 66.3 0.64 « 73 16.6 13.3 17 -u Xt H CO s o o X! Kl 13 -Bank Voltage = 14 KV © Experimental P o i n t s assuming p^ = 1 atmosphere 9.95 23 21 X S h i f t e d P o i n t s S h i f t p' = Assumea D r i v e r ^ Pressure i n Atmospheres P 1 = Test Gas Pressure © 19 Theory of Chapter 2 11 -0 1 2 3 4 l o g 1 Q ( p ^ / p 1 ) Figure 5-5 Measured Shock Speed versus log^ 0(p^/p^) Figure 5-6 Measured Shock Speed versus l o g (p'/p,) 75 l o g 1 0 ( p 4 / p 1 ) J i g u r e 5-7 Measured Shock Speed versus l o g 1 Q ( p ^ / p 1 ) 76 The values of and p^ are seen to d i f f e r by large amounts. Since the theory for the diaphragm opening has been v e r i f i e d under predetermined driver conditions, i t seems l i k e l y that the diaphragm opening times should give r e l i a b l e values for the driver pressure, p^. The fact that the e f f e c t i v e d r i v i n g pressure, p^, i s considerably less than p^ i s probably due to the three-dimensional flow pattern of the driver gas as i t passes through the diaphragm opening. Consider a snapshot of the flow taken at a time when the diaphragm has opened through a small angle 6 (Figure 5-8). The arrows indicate the di r e c t i o n of gas flow; It i s evident that the driver gas Figure 5-8 I n i t i a l Driver Gas Flow Pattern i n i t i a l l y expelled through the small opening w i l l have a large r a d i a l component of momentum. The quantity of driver gas which passes through the diaphragm during the opening process and which consequently executes th i s r a d i a l motion can be estimated from the mean diaphragm aperture during the opening process and the expulsion speed of the driver gas. The l a t t e r can be calculated from the shock speed near the diaphragm. Such a c a l c u l a t i o n shows that a large f r a c t i o n of the driver gas i s expelled during the opening time and must be affected by-three-dimensional motion. The energy involved i n this r a d i a l motion i s probably l a r g e l y absorbed by the shock tube walls and so i s i n e f f e c t i v e i n driving the shock. Consequently, the a x i a l momentum transfer to the test gas i s much less than i n the id e a l s i t u a t i o n , where only a x i a l flow occurs. The shocks should behave then as though driven by a considerably reduced pressure, as we observe. The results presented i n the table above show that p^ and a^ are related by Pj = B ( a | ) 2 . (5.4) - 2 -3 where B is a constant equal to 8.1(±0.8)*10 Kg-m T h e o r e t i c a l l y , one would expect that V\ - - J - Cap 2 (S.S) where and ^ are the mass density and adiabatic exponent of the driver gas. Assuming ^ = 1.67, one finds fit- = 9.8-10"2Kg.m"3, in reasonably good agreement with the value of B found experimentally. It therefore seems, i n spite of the i n i t i a l three-dimensional d r i v e r gas flow pattern, that the i d e a l one-dimensional theory can be applied to a good approximation provided we use an e f f e c t i v e driver gas pressure, p^, and an 1/2 e f f e c t i v e driver gas sound speed, a| = ( 3^4P4/ JP 4^  Empirically we find that >4 PA » ( P J S C5.6) 78 for 15 atm. < p 4 < 70 atm., where s = 0.63 ± 6%, and p 4 and p^ are expressed i n atmospheres. It i s now possible, using equation C4.8) for p^ and the above relations for p| and a^, to predict shock speeds i n the Smy shock tube by means of the standard shock tube theory knowing only the capacitor bank voltage, V ,' the test gas pressure, p^, and the driver gas mass density, J>^> which may be calculated from the driver f i l l i n g pressure since the driver gas i s heated at constant volume. It has s i m i l a r l y been observed that even for cold-driver diaphragm shock tubes the i d e a l theory does not apply and that an e f f e c t i v e d r i v i n g pressure has to be introduced to predict the Mach numbers. This discrepancy has been attributed to a v a r i a t i o n of $^ (the adiabatic exponent of the driver gas) with temperature as the driver gas cools through the expansion wave (see Section 2.4). We f e e l that this explanation does not apply in our case since the )f for helium i s almost constant for a l l temperatures below about 1.5 • l O ^ K ^ 3 5 ^ , and since the driver gas temperature i s less than this value when the diaphragm breaks for bank voltages less than 12 KV. The three-dimensional motion of the expanding driver gas i s then probably responsible for the discrepancies we observe. It i s inte r e s t i n g to note that the exponent i n equation (5.6) i s very nearly ^ - 1 for the helium driver gas. This fact suggests that one should try a polyatomic driver gas with a low value of Jf4 to see i f the exponent i s indeed If. - 1. 79 5.4 The E f f i c i e n c y of the Shock: Tube The f r a c t i o n of the i n i t i a l bank energy dissipated i n the driver gas was measured as follows. A Tektronix P6013 1000:1 voltage d i v i d e r was used to measure the voltage across the driver electrodes, . The discharge current, I , was also measured under the same conditions. The power d i s s i p a t i o n , I ' V J J was calculated and plotted as a function of time. The area under the graph was measured with a planimeter to give the energy d i s s i p a t i o n , W = J'l'V^dt. The e l e c t r i c a l e f f i c i e n c y i s defined as £ C V C 2 CS.7) and the results are given i n Table I I . V c (KV) 1/2CV2 c (Joules) W (Joules) 7 1225 610 501 10 2550 1360 54% 14 4900 2540 51% Table I I E l e c t r i c a l E f f i c i e n c y of the Shock Tube An e l e c t r i c a l e f f i c i e n c y of about 50% i s expected since the bank energy should be divided roughly equally between the dri v e r spark and the a i r trigger spark. Since t h i s i n e f f i c i e n c y can be larg e l y avoided by a l t e r i n g the design of the d r i v e r section to include a triggering f a c i l i t y , i t w i l l not be further considered. 80 Assuming that the energy, W, a l l appears as thermal energy of p a r t i c l e motion i n the driver gas, one can calculate an i d e a l pressure, p^ -, for the driver gas. A thermal e f f i c i e n c y , 7|T, can then be defined as ^ T - P 4/P T (5.8) where pj = W/(Volume of driver gas). and p 4 i s the driver pressure determined from the diaphragm opening time. Results of these calculations are presented'in 3 ,Table III for the driver gas volume of 167 cm i n i t i a l l y at atmospheric pressure and room temperature. V c (KV) Pi (atm) P4 (atm) 7 36.2 16.9 471 10 80.5 33.2 411 14 151 66.3 44% Table III Thermal E f f i c i e n c y of the Shock Tube Thus, about 45% of the energy dissipated i n the driver i s actually e f f e c t i v e i n r a i s i n g the driver pressure. The rest of the energy may be l o s t i n evaporating electrode and wall materials and i n heating the electrodes which have a large surface area. It therefore appears that about 25% of the e l e c t r i c a l energy stored i n the capacitor bank i s e f f e c t i v e in heating the driver gas to a pressure p.. CHAPTER 6 THE SHOCK-HEATED GAS 6.1 Introduction The usefulness of a shock tube i s dependent upon i t s a b i l i t y to produce a uniform sample of hot gas v/ith properties calculable from the i n i t i a l test gas conditions and the shock speed. Experience has shown, i n the case of electromagnetically driven shock tubes, that a cursory examination of the shock speed i s i n s u f f i c i e n t to esta b l i s h these c o n d i t i o n s t 3 * 4 ^ . In the development of a shock tube, i t is therefore e s s e n t i a l to sc r u t i n i z e i t s performance i n d e t a i l . In Chapters 4 and 5, we have shown that the shock speed i n the Smy shock tube can be predicted from the independent i n i t i a l parameters, p^, p 4, and V"c. We now turn our attention to the most important question of the quantity and properties of the shock-heated gas. Frequent reference w i l l be made to the regions of gas flow i n the shock tube. These are i l l u s t r a t e d i n Figure 6-1, and a description of them is given i n Chapter 2. The region of gas between the shock front and the contact surface i s ca l l e d the shock-heated gas even though i t s properties have not yet been shown to conform to the predictions of shock wave theory. Figure 6-2 i s a smear camera picture taken with the s l i t perpendicular to the shock tube ax i s . The shock front, relaxation zone, shock-heated region, and contact zone are c l e a r l y seen. (In p r a c t i c e , the test and driver gases are not sharply separated but tend to mix with each other over an Wall Boundary Layer Test Shock-Heated D r i v e r Gas Gas Gas 1 2 3 X.^---—— Shock Onset of Contact Zone Front E q u i l i b r i u m Conditions R e l a x a t i o n Zone Figure 6-1 A x i a l Section of Shock Tube Showing Regions of Gas Flow |L J„ Shock Shock-Heated Contact Zone Front Gas H o r i z o n t a l Scale: 1 cm. on f i l m =3.6 cm. i n shock tube Figure 6-2 Smear P i c t u r e of Shock Taken with . S l i t Perpendicular to Shock Tube A x i s (p 1=10 Torr, V c=10 KV, x=120 cm) 83 extended region which we c a l l the contact zone. The d i f f u s e , somewhat i r r e g u l a r nature of t h i s zone i s evident i n Figure 6-2. The leading edge of th i s zone we c a l l the contact surface.) The contact zone was i d e n t i f i e d by a shock r e f l e c t i o n technique which i s described in the next section. The luminous l i n e seen at the shock front i s presumably due to e a s i l y excited impurities for which the relaxation times are s h o r t ( 1 0 > 1 4 » 1 9 » 2 9 » 5 2 ) # T h i s luminous l i n e shows that the shock front i s planar over at lea s t the central 801 of the tube diameter. In Section 6.2, the quantity of shock-heated gas i s investigated and accounted for i n terms of a simple model. The properties of this gas are discussed i n Section 6.4. 6.2 Extent of the Region of Shock-Heated Gas Ideally, for a shock wave of constant speed, V , and density r a t i o , ^/f{ > t n e region of shock-heated gas should have a thickness ;'-ltw = * P / f 2 (6.1) at a distance x from the diaphragm. Also, the speed of the contact surface, V , should be given by V « = V , ( l - % ) (6.2) These equations follow from the requirement of mass conservation. From (6.2) the thickness of shock-heated gas should increase at the i d e a l rate _ \/ _ w _ \/ A d t - V S - V C 5 = V . - £ - (-3) as the shock travels down the tube. In deriving these expressions, i t i s assumed that a l l of the test gas encountered by the shock i s compressed into a layer between the shock front and contact surface, and follows the shock at a speed V . If thi s assumption were v a l i d , then a measurement of the shock-heated gas thickness or of the r e l a t i v e v e l o c i t i e s of shock front and contact surface would serve to determine the shock density r a t i o , In r e a l i t y , however, loss mechanisms reduce the thickness of the shock-heated layer. The mixing of test and driver gases i n the contact zone e f f e c t i v e l y advances the contact surface (the leading edge of this zone) towards the shock f r o n t . Also, the establishment of a viscous boundary layer at the shock tube wall requires that gas near the wall have a low v e l o c i t y . It can then be passed over by the contact surface. Furthermore, the temperature of gas near the wall must be that of the w a l l . Since i t is assumed that no pressure gradient exists across the boundary layer, the gas density near the wall can be much higher (311 than near the axis'- ' . A large f r a c t i o n of the shock-heated gas can thus be removed by the boundary layer. These processes reduce the thickness of shock-heated gas and increase the v e l o c i t y of the contact surface. Consequently, the observed thickness, l m , of shock-heated gas gives an apparent density r a t i o , RL, such that l m - X/RL (6.4) i f the losses are not taken into consideration. 1 w i l l be less than 1^ and R L correspondingly greater than the t h e o r e t i c a l value, P2^/p> . In a s i m i l a r way, a measurement of the contact surface speed, VQSm and shock speed, V g, gives an 85 apparent density r a t i o , R^, related to the rate of increase of 1 m csm (6.5) Now, due to the loss of shock-heated gas, V w i l l be greater the instantaneous rate of l o s s . Although the loss of gas through the boundary layer and through mixing reduces l m , these processes do not af f e c t the properties of the shock-heated gas and so can be tolerated provided that a l l of the gas i s not removed. may affe c t the observed length of shock-heated gas and i t s rate of increase. When the diaphragm breaks and driver gas rushes downstream, a rarefac t i o n wave travels towards the closed end of the driver section where i t i s r e f l e c t e d . After r e f l e c t i o n , the r a r e f a c t i o n wave proceeds downstream i n pursuit of the shock front at a speed given by the sunt of the l o c a l flow v e l o c i t y and the l o c a l speed of sound. In region 5 (see Figure 6-1) behind the contact surface, t h i s wave has a speed v^ + a^. Since the contact surface has a speed v^, i t w i l l c l e a r l y be overtaken by the r a r e f a c t i o n wave at some point in a s u f f i c i e n t l y long shock tube. The point at which this occurs depends on the i n i t i a l conditions, the shock speed and the length of the driver section. The contact surface w i l l then be retarded. As a r e s u l t , the shock-heated layer w i l l increase i n thickness at a rate greater In addition to the loss mechanisms, another phenomenon 86 than predicted by (6.3) , and the apparent density r a t i o , R^ ., w i l l f a l l below the th e o r e t i c a l value, &/p » Also, the ra r e f a c t i o n wave w i l l pass into the shock-heated gas C r e g i ° n 2 of Figure 6-1) and a l t e r the properties of thi s gas. Clearly t h i s e f f e c t i s undesirable and should be investigated i n examining the merits of a shock tube. A comparison of the measured values of and with the i d e a l density r a t i o , *^a/p » should then give an ind i c a t i o n of the importance of loss mechanisms and of the e f f e c t of the rar e f a c t i o n wave. To investigate the extent and growth of the shock-heated region a six-foot length of pyrex pipe was used i n the shock tube. The gas flow between positions 100 cm and 130 cm from the diaphragm was observed with the smear camera. The photo-graphs (Figure 6-3) present x-t diagrams of the luminous features of the flow in the shock tube. The oblique "streaks" on the f i l m representing the progress of these features w i l l be c a l l e d t r a j e c t o r i e s . The v e r t i c a l black lines are markers spaced at 10 cm interv a l s along the shock tube. In order to i d e n t i f y the contact surface and, at the same time, to render weak, non-luminous shocks v i s i b l e , the gas flow was re f l e c t e d from a l u c i t e d i s c f i t t e d f l u s h with the inside surface of the shock tube and perpendicular to i t s axis at a p o s i t i o n 123 cm. downstream. The trajectory of the contact surface could be i d e n t i f i e d by the change i n speed, or r e f r a c t i o n , of the (2 30") r e f l e c t e d shock as i t passed through the contact zone*- * J . To reduce the in t e n s i t y of the l i g h t emitted by the doubly-N e u t r a l Density F i l t e r Covers Upper Part of F i l m (a) Test Gas Pressure (b) p1 = 0.5 Torr V: 15 jisec/cm V: Denotes V e r t i c a l (Time) Scale on F i l m Location of N e u t r a l Density F i l t e r Figure 6-3 Smear Photographs of Gas Flow i n Shock Tube H o r i z o n t a l Scale: 1 cm on f i l m = 5.75 cm i n shock tube a x i a l d i r e c t i o n (c) p 1 = 1 Torr (d) J91 = 2 Torr V: 15 (isec/cm V: 15 u.sec/cm =7.5 Torr 30 (j.sec/cm (h) p = 10 Torr V: 30 usec/cm 90 + p "j P o s i t i o n of x Thickness of R e f l e c t i n g Shock-Heated Gas P l a t e Regions 1 Test Gas i n I n i t i a l State of 2 Shock-Heated Test Gas Gas 3 D r i v e r - T e s t Gas Mixture Flow 5 Doubly-Shocked Test Gas Figure 6-4 E x p l a n a t i o n of the Features of the Smear Camera Photographs of Figure 6-3 91 shocked gas (region 5 of Figure 6-4) i t was necessary to place a neutral density f i l t e r diagonally across the photographic f i l m as shown i n Figure 6-3a. This explains the abrupt "fading" of the luminous features associated with the incident shock as they approach the r e f l e c t i n g p l a t e . A l i n e diagram (Figure 6-4) explains the relevant features of a t y p i c a l smear picture. The sharpness of the shock r e f r a c t i o n suggests that the contact zone i s f a i r l y t h i n . Colour smear pictures also exhibit a marked colour change i n the contact region. It appears there-fore that a well-defined region of shock-heated gas e x i s t s . From the smear pictures i t i s then possible to measure the thickness of shock-heated gas, l m , (Figure 6-5), as well as the v e l o c i t i e s of both the shock front and the contact surface. The smear pictures also show that for < 5 Torr, the length of the relaxation zone i s small compared with 1^. The importance of this condition i s discussed i n Section 2.6. A l l measurements were made with constant driver con-diti o n s . The shock speed was varied by changing the test gas pressure, p^. The dependence of the shock speed on p^ for the driver conditions used (V = 10 KV) i s shown i n Figure 6-5. The results are presented below and then an attempt i s made to account for them q u a n t i t a t i v e l y . In Figure 6-6, the r a t i o , Ijj/lj-h* of measured to th e o r e t i c a l thickness of shock-heated gas i s plotted as a function of downstream pressure, p^. In a l l cases, t h i s r a t i o • i s less than unity. Two possible reasons for this deviation e x i s t . Either the density r a t i o across the shock i s not the 92 20 18 16 14 M 12 -10 -8 P 1 (Torr) Fi g u r e 6-5 Dependence of Shock Mach Number (M) and Shock-Heated Gas Thickness ( l m ) on Test Gas Pressure (p-^) 1.0 0.8J in "th 0.6 0.4 0.2 0 0 Bank Voltage = 10 KV Measuring Station 120 cm. from Diaphragm Theory (Equations (6.13) and (6,16)) 0 Experiment i 8 P x (Torr) —r 10 12 14 Figure 6-6 Comparison of Measured Thickness of Shock-Heated Gas ( l m ) with Ideal Value ( l t h ) C D • ' w value, P 2 / P ] , »' determined from the shock speed (and used i n ca l c u l a t i n g 1 ^ ) , or some of the test gas has been lost from the shock-heated layer. Now, i n view of experiments described i n Section 6.4, i t appears that the shock density r a t i o agrees with the calculated value. The difference between 1 and 1., must • . • m t n then be due to gas lo s s . The measured contact surface speed, V" . i s plotted r ' csm* r against the measured shock speed, V , i n Figure 6-7. Also shown are the l i n e V = V and the th e o r e t i c a l dependence of V" on csm s r cs V . The measurements are seen to d i f f e r from the predicted s values for both high shock speeds (low p^) and low shock speeds (high p-.). The approach of V to V i s attributed to loss of gas from the shock-heated layer since i t seems u n l i k e l y that the density r a t i o should d i f f e r much from the t h e o r e t i c a l value. The discrepancy for low V s ( i . e . p.^  greater than about 4 Torr) i s probably due to the reflected' r a r e f a c t i o n wave overtaking and retarding the contact surface. Evidence i n support of thi s explanation i s contained i n some smear pictures taken for greater than 5 Torr. In some of these photographs, subsequent t r a j e c t o r i e s in region 3 (Figure 6-4) have increasing slopes and therefore represent flow of decreasing speed. A rar e f a c t i o n wave advancing towards the contact surface would produce th i s e f f e c t . The observed density ra t i o s R L and Ry, obtained from measurements of the thickness of shock-heated gas and i t s rate of increase respectively (equations (6.4) and (6.5)), are com-pared with the ide a l value, p 2/p,, i n Figure 6-8. Over the Bank Voltage » 10 KV Measuring S t a t i o n 120 cm. from Diaphragm «5 Figure 6 - 7 Contact Surface Speed (V ) as a Function of Shock Speed (V s) o •rl +-> ei >> 4-> •rl ID a Q 25' 20-15-10-B a n k V o l t a g e = 10 KV M e a s u r i n g S t a t i o n 120 cm. f r o m D i a p h r a g m P, T h e o r e t i c a l D e n s i t y R a t i o ( C a l c u l a t e d f r o m M e a s u r e d S h o c k S p e e d ) A p p a r e n t D e n s i t y R a t i o ( C a l c u l a t e d f r o m L e n g t h o f S h o c k - H e a t e d Gas L a y e r ) A p p a r e n t D e n s i t y R a t i o ( C a l c u a t e d f r o m R a t e o f I n c r e a s e o f S h o c k - H e a t e d Gas T h i c k n e s s ) to 0-5 1 1 1 1 ' ' 1 ' 0 2 4 .6 8 10 12 14 16 T e s t Gas P r e s s u r e ( T o r r ) F i g u r e 6-8 C o m p a r i s o n o f D e n s i t y R a t i o s whole range of pressures investigated, ML > 9x/pl C6.6) Assuming that the shock-heated gas kas density, p 2, this r e s u l t implies that some of the downstream gas encountered by the shock front has been lo s t and does not appear i n the shock-heated layer. Furthermore, the graph also shows that for a l l pressures used - R L > Ry (6.7) Hence, gas i s lo s t early i n the f l i g h t of the shock at a rate greater than the loss rate at the observation s t a t i o n 120 cm from the diaphragm. The i n i t i a l high loss of gas i s probably due to mixing of test and driver gases during the formation of the shock front. A further feature of.Figure 6-8 deserves comment before a model for the gas flow is discussed. Two flow regimes can be distinguished by the r e l a t i v e sizes of Ry and • For less than about 4 Torr, Ry > Pa ,^ . As explained in discussing equation (6.5), this results from the loss of test gas, probably through a viscous boundary layer. For p^ > 4 Torr, Ry < ij/j? ^ i n d i c a t i n g that the thickness of shock-heated gas i s increasing at a rate greater than can be accounted for on the basis of i d e a l shock tube theory. The retardation of the contact surface i n th i s pressure range has already been attributed to the r e f l e c t e d rarefaction wave. The two flow regimes w i l l be referred to as the "boundary layer l i m i t e d " regime (p^ < 4 Torr) and the "rarefaction wave li m i t e d " regime (p, > 4 Torr). 98 In formulating a model to explain the observed length of shock-heated gas, 1 , the "boundary layer l i m i t e d " regime w i l l be treated f i r s t . We s h a l l assume that a length of test gas mixes with the driver gas and therefore cannot contribute to 1 . Since Hooker^ 2^ has shown that the length of the column of mixed dr i v e r and test gases i s independent of downstream pressure, we are j u s t i f i e d i n assuming that does not depend on p^. We also assume that of the available gas downstream of only a f r a c t i o n ^ , ( x jPi) 1 S retained, the remainder being lost through a viscous boundary layer. Conservation of mass then requires that the length of the shock-heated gas behind a shock at x Q i n the tube i s given by x A1* = -ft / M X (6.8) Since we do not know the dependence of ^ on x we s h a l l evaluate the i n t e g r a l approximately.„-We assume that a l l the test gas o r i g i n a l l y between and some p o s i t i o n L 2 remains in the shock-heated layer, so that /5, = 1 for •£ x i L>2. For L 2 < x 4 x Q, a constant f r a c t i o n 9^0(p,) *= 1 of the test gas i s assumed to be retained, the remainder being l o s t through a viscous boundary layer. We define 9^0(f>,) = (i, (*o,f/) . Since we expect the loss of gas to increase with x (or equivalently, with the separation of shock and contact surface), .fl(X,p,) I /30CP.) = A<X,P.) for X * Xo (6.9) We further assume that L 2 i s independent of downstream pressure. With these assumptions (6.8) becomes P j « = / 5 M *o - l 2 ) * JJCU-L.) ( 6 > 1 0 , 99 . for p-^  < 4 Torr. By d i f f e r e n t i a t i o n , ^ D i s found to be r l 1° j> dt UtJX=; or where the v e l o c i t i e s are measured at x Q. That i s , fiQ i s the r a t i o of the ide a l density r a t i o to the apparent density r a t i o determined from the r e l a t i v e v e l o c i t i e s of shock and contact surface at x Q. From equation (6.5), jfe ( 6 , 1 2 ) which can be evaluated from Figure 6-8. P lm Equation (6.10) predicts that a graph of — J 1 a~ against P P o should be a straight l i n e for p^ < 4 Torr, .with slope ( L 2 - L 1) and intercept (x Q - L 2 ) . Figure 6-9 shows that this i s indeed the case and gives-L, = 56 cm, L 0 = 89 cm, for x = 120 cm 1 ' 2 o Two conditions under which the in t e g r a l of (6.8) can be written as i n (6.10) f o r constant L 2 are shown i n Figure 6-10. There the normalized instantaneous gas loss i s p lotted as a function of distance x, where the f r a c t i o n a l gas loss i s defined as SO^P. ) = I - / 3 ( ( X j P ( ) (6.14) and f0Cp,) = • i - /S0Cp.> 0.5 40 i 1 r 1 1 1 « ! f 0.4 0.6 0.8 1.0 , 1.2 1.4 1.6 1.8 2.0 B F i g u r e 6-9 Experimental Check on the V a l i d i t y of Equation (6.10) 101 (a) S ( x , P i ) l s a separable f u n c t i o n of x and p^ or approximately soj that i s , ^(XjP^) changes l i t t l e f o r a l a r g e change i n p^. Figure 6-10 Two S i t u a t i o n s i n which the I n t e g r a l of Equation (6.8) may be approximated as i n Equation (6.10). 102 In case ( a ) , the more l i k e l y s i t u a t i o n , G(x,p) e i t h e r i s independent of p^ or depends v e r y weakly on p^ over the whole range of < x < x ; i-e-" Gu,p.) = g i * ; ? , \ = n*) (6.15) 5 o ^ P.) where 7^  depends only on x. Then J^Cx^f?) i s a sep a r a b l e f u n c t i o n of x and P j : ' From (6.9) _, . ' , . „ , , ?(X,P,) £ f 0(p,) for * -5 0 t h a t Oi 7J(X) .6 I C 6 ' 1 7 ) We can r e w r i t e (6.8) i n terms of C ( ( * J P I ) as f o l l o w s : f j m = P(CXo-L,) - J>,/ ?(*>P,) d x (6.18) From (6.16) ^()C)dx (6.19) L. Now the i n t e g r a l must have dimensions of l e n g t h . We t h e r e f o r e d e f i n e such t h a t .X0 J >9U)cU = X 0 " L z (6.20) where, from (6.17) x„ - L 0 < x„ - L, ; i . e . L 9 > L, o 2 o 1 L l Equ a t i o n (6.18) then becomes ?S = J^^0"1') " f, ? O ( P , ) ( X O - LJ ( 6 > 2 1 ) Upon s u b s t i t u t i n g foCp,) = l-/30(p,) , equation (6.21) becomes i d e n t i c a l t o (6.10). From (6.20), L 2 i s c l e a r l y independent o f p ^ 103. In case (b) GCx,p1) may depend strongly on p^, provided the curves for various p^ d i f f e r only over small intervals of t o t a l length A << x - L^. ^ ( X j p ^ ) w i l l then be separable over most of the region between and X q , ; and (6.10) w i l l again hold approximately for I^ independent of p^. For p^ greater than about 4 Torr, the graph of Figure 6-9 becomes a horizontal l i n e . We seek an i n t e r p r e t a t i o n of this phenomenon in terms of the r a r e f a c t i o n wave which enters the region of shock-heated gas and affects the flow i n t h i s pressure range. As Figure 6^ 8 shows, |3 0 approaches unity as p^ approaches 4 Torr. We s h a l l therefore assume that boundary layers are n e g l i g i b l e for p^ greater than 4 Torr. Now, due to the presence of the r e f l e c t e d r a r e f a c t i o n wave i n the shock-heated gas, we would expect the density of t h i s region to be non-uniform and to have an average value JOz. , where (b > 1 As before, a constant length, , of t e s t gas i s assumed to be removed by i n i t i a l mixing processes. In t h i s " rarefaction wave l i m i t e d " regime, then, the shock-heated gas layer has a thickness, 1 , given by 4 ^ = P,(*o-L.) (6.22) This equation i s of the same form as (6.10) and ^ may be found as before by d i f f e r e n t i a t i n g : As already pointed out, however, the interpretations of fit and /32 d i f f e r . Equation (6.2 2) states that ^ should be n-1 ' ^ independent of p 2 for p^ greater than 4 Torr and therefore 104 successfully accounts; for the horizontal portion of the graph i n Figure 6-9. Moreover, the intercept, x Q - L^, gives L 1 = 5.9 cm (6.24) i n good agreement with the value of = 56 cm obtained from the "boundary layer l i m i t e d " flow regime (see page 99). In th i s sense, then, the equations describing the two flow regimes are consistent. Substituting (6.13) into (6.10) and (6.22) y i e l d s = 0.259 3 + 0.275 , p, < 4 Torr 1 o l xt h (6.25) - S - = 0.508 B 2 , p x > 4 Torr 1 t h Using the experimental values of 8 and 8 7, these equations are plo t t e d i n Figure 6-6. The agreement with the experimentally determined values of 1 ^ 1 ^ i s seen to be very good. In spite of i t s crude nature the model therefore seems to explain the observations. In summary, i t seems that we can explain the observed thickness of shock-heated gas by a simple physical model. Two regimes of flow are apparent according as the downstream pressure, p^, i s less than or greater than about 4 Torr. In both cases, a constant i n i t i a l length, L^, of test gas is assumed to mix with the driver gas. For low pressures, test gas between and a p o s i t i o n L 2 , independent of downstream pressure, i s assumed to c o l l e c t without loss between the shock front and contact surface. Between L 0 and the observation s t a t i o n at x , L o a viscous boundary layer then i s assumed to remove a constant 105 f r a c t i o n of the gas encountered by the shock f r o n t . In the high pressure regime, the flow i s assumed to be i n v i s c i d . After the i n i t i a l mixing over length L^, the shock-heated layer i s assumed to r e t a i n a l l the test gas entering i t . The overtaking of the contact surface by the r e f l e c t e d r a r e f a c t i o n wave i s invoked to . \ explain the length and growth rate of this layer. It should be emphasized that when thi s occurs the properties of the shock-heated gas are altered and the shock tube loses i t s usefulness. The model predicts a length of shock-heated gas consistent with the observed value for i n the range 0.5 to 15 Torr. For d i f f e r e n t driver conditions or observation stations (we used V"c = 10 KV and X q = 120 cm) , the c r i t i c a l pressure separating the two flow regimes (4 Torr i n our case) i s expected to change. 6.3 Currents i n the Shock-Heated Gas In some electromagnetically driven shock tubes, discharge currents have been detected i n the shocked gas immediately f3 4*) behind the shock f r o n t v * J . Such currents a l t e r the conditions of the shock-heated gas and render i n v a l i d the ap p l i c a t i o n of standard shock wave theory. The experiments of Chapter 4 i n d i -cate that the behaviour of shock waves produced i n the present shock tube i s decoupled from the active stage of the e l e c t r i c a l discharge since the diaphragm does not begin to break u n t i l the discharge current has f a l l e n to a small f r a c t i o n of i t s peak value. (See e.g. Figure 4-11. For V c = 10 KV and diaphragms 0.005 inch thick, the diaphragm breaks when the current has f a l l e n to about 10% of i t s peak value.) Also, a f t e r the 106 diaphragm does break, the current decays to an undetectably 3 small value (< 10 amps) within a few microseconds. It was f e l t expedient, however, to examine the shock-heated gas for small currents. ;>: For this purpose, a small 100-turn c o i l 5 mm long and 2.5 mm i n diameter was inserted into the shock tube at a p o s i t i o n 136 cm from the diaphragm. The signal was amplified d i f f e r e n t i a l l y and displayed on a dual-beam oscilloscope together with the signal from a pressure probe located opposite the c o i l at the same a x i a l p o s i t i o n . Observations were made with the c o i l axis aligned along three mutually perpendicular axes. Results are shown in Figure 6-11. Due to the acoustic delay i n the pressure probe, i t s signal i s delayed 18 micro-seconds with respect to the shock a r r i v a l . The c o i l signals appear roughly the same for a l l orientations of the c o i l . This fact suggests that the signal may be caused by capacitative coupling between the shock-heated plasma and the c o i l . Furthermore, the fact that the signal disappears within 30 microseconds i n a l l cases strongly indicates that the signal i s not due to a current associated with the discharge since such a current would p e r s i s t i n the d r i v e r gas as well. However, admitting such a current and assuming that i t goes, undetected in the d r i v e r gas for some reason, the c o i l s ignal amplitude places an upper l i m i t of about 2 amps on the current. Since the plasma conductivity i s f 3 3 1 r e l a t i v e l y high under the conditions used (Williamson v v 3 -1 estimates i t to be about 4'10 mho-m ), heating produced by Shock Tube C e i l H o r i z o n t a l P a r a l l e l to Tube Axis Probe C o i l C o i l V e r t i c a l Perpendicular to Tube Axis C o i l H o r i z o n t a l Perpendicular to Tube A x i s Figure 6-11 Attempts to Observe Currents i n Shock-Heated Gas Behind Mach 16 Shock i n Argon a t 1 Torr Upper Beam: Pressure Probe 0.2 V / d i v i s i o n Lower Beam: Search C o i l 0.5 V / d i v i s i o n Time (Right to L e f t ) : 20 u s e c / d i v i s i o n 108 ohmic d i s s i p a t i o n of thi s current i s much smaller than the energy added by shock wave heating. The current - i f there is i n fact a current in the shock-heated gas - w i l l then have n e g l i g i b l e e f f e c t on the properties of the shock-heated gas. 6.4 Properties of the Shock-Heated Gas In section 6.2, the measured thickness of shock-heated gas was shown to increase at less than the predicted rate for pressures below about 4 Torr. The apparent density r a t i o across the shock front was therefore greater than expected. I f these effects are attri b u t a b l e to leakage of gas past the contact surface v i a a viscous boundary layer, then one would s t i l l expect the thermodynamic properties of the shock-heated gas to be the predicted values. In this section these properties are investigated. (a) Pressure Behind the Shock Front - From equations (2.1) and (2.5) i t follows that for a strong shock wave (p 2 >> p-^ , and u^ >> u 2) the pressure, p 2 , behind the shock i s given to a good approximation by P 2 X ftVg (6.26) where is the test gas mass density and V g i s the shock speed in the laboratory. V g and p^ can be measured and a signal of amplitude A, proportional to the shock-heated gas pressure, p 2 , can be obtained from a pressure probe. Since i s proportional to the test gas pressure, p^, (6.26) predicts that A oc p.V? (6.27) 109 The measured amplitude, A, o f a pressure probe s i g n a l i s p l o t t e d 2 i n F i g u r e 6-12 as a f u n c t i o n of the measured value of P-jVs« • 2 i s seen t h a t A i s p r o p o r t i o n a l to P ^ v g , except f o r l a r g e v a l u e s o f the l a t t e r q u a n t i t y , f o r which l a r g e v a l u e s of p 2 are expected and the probe response i s p r o b a b l y n o n - l i n e a r . Since the p r e s s u r e , p 2 , has the expected l i n e a r dependence 2 on P j V g , i t seems very l i k e l y t h a t p 2 conforms t o the p r e d i c t i o n s o f Rankine-Hugoniot t h e o r y . One should attempt an absolute p r e s s u r e measurement to prove t h i s c o n c l u s i v e l y , but an accurate technique f o r c a l i b r a t i n g the p r e s s u r e probe, such as t h a t (391 developed by Katsaros- , was not a v a i l a b l e , (b) Sound Speed i n the Shock-Heated Gas The purpose of t h i s experiment was to measure the temperature of the shock-heated gas by o b s e r v i n g the speed of sound i n the gas. The measured temperature c o u l d then be com-pared w i t h the v a l u e p r e d i c t e d by the Augmented Rankine-Hugoniot theory of Chapter 2. For. a shock-heated gas A h l b o r n ^ ^ has shown t h a t the <32 = •23kI± (6.28) sound speed, a 2 , i s given approximately by Here g 2 i s the e f f e c t i v e a d i a b a t i c exponent of the shock-heated gas g i v e n by h 2 u 2 where u 2 and h 2 are the s p e c i f i c i n t e r n a l energy and s p e c i f i c e n t h a l p y , r e s p e c t i v e l y . Graphs of g 2 as a f u n c t i o n of (351 temperature are given, by Ahlborn and S a l v a t v 1 . The p r e s s u r e , 120 © F i g u r e 6-12 Dependence of Pressure Probe Signal, Amplitude on P-.V2 p 2 , can be expressed as assuming the shock-heated gas to be i n thermal e q u i l i b r i u m at temperature, T 2 . Here e\ i s the degree of i o n i z a t i o n and n 2 i s the heavy p a r t i c l e number d e n s i t y (atoms and i o n s ) . The mass d e n s i t y i s g i v e n by fx = " 2 ™ (6.30) where m i s the mass of an atom or i o n . The c o n t r i b u t i o n to p from f r e e e l e c t r o n s has been ig n o r e d . E q u a t i o n (6.23) then becomes a* = ^ — — — (6.3i) I f i o n i z a t i o n i s n e g l i g i b l e (low temperatures), then, f o r a monatomic gas, g 2 becomes the f a m i l i a r r a t i o of s p e c i f i c h e a t s , $ , and we get the u s u a l e x p r e s s i o n f o r the sound speed a * = ~ m — ( 6 , 3 2 ) Even f o r a s m a l l degree of i o n i z a t i o n equation (6.32) g i v e s a f a i r l y good estimate o f . t h e sound speed s i n c e the i n c r e a s e of C< and decrease of g 2 compensate each other somewhat as T 2 i n c r e a s e s . Hence from a measurement of a 2 , T 2 can be obtained approximately. To measure a 2 , a p r e s s u r e d i s t u r b a n c e was produced i n the shock-heated gas and the speed of t h i s p u l s e was measured w i t h a t i m e - o f - f l i g h t technique u s i n g a p r e s s u r e probe to d e t e c t the p u l s e . E l e c t r i c a l energy from a 0.25 y f a r a d c a p a c i t o r was d i s c h a r g e d through a t i n y spark gap (Figure 6-13) T r i g g e r a b l e A i r Spark Gap Switch Capacitor 0.25 i i f a r a d I n s u l a t e d Copper S t r i p s Clamped Together C i r c u i t Parameters: Rin g i n g P e r i o d = 3 |isec 5 _ i Logarithmic Decrement = 1.7*10 sec Tungsten Wire diameter Set Screw 2 mm. to Figure 6-13 D e t a i l s of Spark Gap Used to Produce "Sound" Pulse 113 on one wall of the shock tube to produce a p-ressure pulse after the shock wave had passed that point i n the shock tube. A pressure probe inserted through the opposite side of the shock tube detected the a r r i v a l of the pulse. (See Figure A-3 i n the Appendix for t y p i c a l pressure probe signals.) Since the pulse propagated through flowing gas, i t was necessary to displace the detecting probe a x i a l l y downstream i n order to observe the pulse, as shown in Figure 6-14. On subsequent shots under identical•conditions the a r r i v a l time of the pulse was measured for d i f f e r e n t detector probe locations in order to determine the sound speed. The pressure pulse amplitude was f i n i t e (and decreased with distance from the source) so one would expect the speed of propagation to be greater than the true sound speed. In order to minimize this e f f e c t , the capacitor charging voltage was reduced u n t i l the pulse was just detectable for a t y p i c a l p o s i t i o n of the probe. A suitable voltage was found to be 4.4«10 3 v o l t s . To determine the r e l a t i o n between the pulse propagation speed and the a r r i v a l time at the probe, consider the s i t u a t i o n of Figure 6-14. The acoustic source i s assumed to radiate hemispherically. The probe then detects that portion of the pressure disturbance which has a net v e l o c i t y given by Vs, = V £ + a 2 (6.33) where v 2 i s the gas flow v e l o c i t y and a 2 i s the v e l o c i t y of the pulse with respect to the gas. In the two-dimensional Shock Front Small Spark Gap' Net Velocity- of "Sound" Pulse S l i d i n g Pressure Probe Mount A J v 2 Gas Flow -Pressure Probe Figu r e 6-14 Arrangement f o r Measuring Sound Speed i n Shock-Heated Gas 115 co-ordinate system of Figure 6-14 (i and * are unit vectors) , Vs, = j - i + - | - J (6.34) where t i s the source-to-probe time of f l i g h t of the pulse. Substituting (6.34) and (6.35) into (6.33) - ^ s (t* - + T" ^ (6-36:> Thus aJtV = ( S - V z t ) + A 2 , (6.37) The pulse speed, a 2 , can then be found by measuring the a r r i v a l 2 2 time, t , for d i f f e r e n t s and A and pl o t t i n g (s - v 2 t ) + A 2 against t . The gas flow speed, v 2 , i s calculated from the Augmented Rankine-Hugoniot equations. This procedure was carried out for a Mach 13 shock in argon at p^ = 1 Torr and the—results are shown in Figure 6-15. The f i n a l part of the graph is seen to be a good straight l i n e . From the slope we fi n d a 2 = 2140 m/sec. However, as already explained, this value i s greater than the true sound speed, and a correction was obtained as follows. The amplitude of the spark-produced disturbance, &p, was observed to be about 201 of the pressure behind the main shock, p 2 . (See e.g. Figure A - 3 ( c ) ( i i ) ) . The Mach number of the disturbance is then found from (2.14) to b e M = [ I F {<*+l>fyfc-U-i)}]Y2 = Lost 3% where P 2/p 2 = 1 + Ap/p 2 = 1.2 i 5%. Dividing the measured disturbance speed by i t s Mach number yields a better estimate of the sound speed, a £ = 1980 m/sec £ 5%. Using (6.32) then gives the gas temperature as T ? = l l l l - 1 0 4 °K S 10%. 116 100 200 2 2 300 t (jisec ) Figure 6-15 Determination of Sound Speed i n Shocked Argon 117 This measured temperature agrees w e l l w i t h the value of 1.08'10^°K p r e d i c t e d by the Augmented Rankine-Hugoniot equations f o r a Mach 13 shock i n argon at 1 To r r . The approximation made i n r e p l a c i n g (6.31) by (6.32) i s qu i t e good since at t h i s temperature 0\, = 0.07 and g 2 i s found i n reference 35 t o be about 1.45. Thus, i n (6.32) a value of & = 1.55 should have been used instead of $ = 1.67. Furthermore, A h l b o r n ^ ^ places a l i m i t of about 121 on the e r r o r of (6.28) from which (6.31) was d e r i v e d , so the use of (6.32) i s j u s t i f i e d i n t h i s case. Within the accuracy of our c a l c u l a t i o n s , then, i t appears that the shock-heated gas temperature conforms to t h e o r e t i c a l p r e d i c t i o n s . This f a c t , together w i t h the study of the shock-heated gas pressure, i s good evidence that the shock-heated gas i s i n a s t a t e of e q u i l i b r i u m determined by the Augmented Rankine-Hugoniot equations of Chapter 2. This con-c l u s i o n a p p l i e s , of course, only i n the region of pressure and shock speed s t u d i e d ( i . e . p^ 52/ 1 T o r r , M W 13). No experiments were performed i n the " r a r e f a c t i o n wave l i m i t e d " flow regime, (c) U n i f o r m i t y of the Shock-Heated Gas The u n i f o r m i t y of the r a d i a t i o n emitted by the shock-heated gas, as seen i n the smear photographs (Figure 6-3), suggests that t h i s region i s one of f a i r l y uniform thermodynamic p r o p e r t i e s . Since the lu m i n o s i t y i s a s e n s i t i v e f u n c t i o n of temperature, any temperature v a r i a t i o n i n t h i s region should produce a n o t i c e a b l e non-uniformity i n the luminous gas. I t was shown i n Chapter 5 (Figure 5-1) that the shock a t t e n u a t i o n i s low (about 20% per meter f o r p, < 5 Torr) . 118 . Low a t t e n u a t i o n i s a requirement f o r the shock-heated gas to be f a i r l y uniform i n the a x i a l d i r e c t i o n , f o r i f the shock speed i s not constant, successive a x i a l samples i n the heated gas w i l l have d i f f e r e n t p r o p e r t i e s since each w i l l have been heated by a shock wave of d i f f e r e n t s t r e n g t h . F i n a l l y , the success of the sound speed measurements suggests that the hot gas p r o p e r t i e s are uniform i n both r a d i a l and a x i a l d i r e c t i o n s and r e p r o d u c i b l e from shot to shot. 119 CHAPTER 7 CONCLUSIONS 1.1 Conclusions In Chapter 1, a number of questions regarding the oper a t i o n of the Smy shock tube were posed. Answers to these questions have been obtained. I t has been demonstrated that the shock tube behaves i n a manner s i m i l a r to c o l d - d r i v e r diaphragm shock tubes. The magnetic f i e l d produced by the "backstrap" has n e g l i g i b l e i n f l u e n c e on the a c c e l e r a t i o n of the d r i v e r gas. The high shock speeds are a t t r i b u t a b l e to the high sound speed and pressure brought about i n the helium d r i v e r gas by the e l e c t r i c a l d ischarge. We have developed a technique to determine the heated d r i v e r gas pressure, p^. From a measurement of the time r e q u i r e d f o r the diaphragm to open completely, t , p^ can be c a l c u l a t e d using a simple, e x p e r i m e n t a l l y - v e r i f i e d model. We f i n d that p 4 = 0.332 V 2 (7.1) f o r 5 KV < V < 15 KV, where V i s the c a p a c i t o r bank voltage i n k i l o v o l t s and p 4 i s i n atmospheres. Although other shock tubes of a s i m i l a r type have been d e v e l o p e d ^ 2 ^ , we have been unable to lo c a t e i n the standard l i t e r a t u r e any reference to u s e f u l measurements of p 4 . Of the e l e c t r i c a l energy stored i n the c a p a c i t o r bank, about 50% i s d i s s i p a t e d i n the discharge plasma, and about h a l f of t h i s energy i s e f f e c t i v e i n r a i s i n g the d r i v e r gas pressure. 120 . I i We have found that standard one-dimensional shock tube theory cannot be a p p l i e d d i r e c t l y to t h i s shock tube f o r the purpose of determining the shock speed from the diaphragm pressure r a t i o , p^/p^. Instead, we f i n d that the shocks behave as i f d r i v e n by an e f f e c t i v e d r i v e r pressure, p^ , l e s s than the a c t u a l d r i v e r p ressure, p 4 , measured from the diaphragm opening time. E m p i r i c a l l y , we f i n d that P4 = C P 4 ) 0 ' 6 3 (7.2) f o r 15 atm. < p 4 < 70 atm., where p 4 and p^ are expressed i n atmospheres, and that where a^ i s the e f f e c t i v e sound speed of the d r i v e r gas, and J>^ i s the d r i v e r gas mass de n s i t y which can be c a l c u l a t e d from the i n i t i a l helium f i l l i n g pressure since the gas i s heated at constant volume. We a t t r i b u t e the discrepancy between a c t u a l and e f f e c t i v e d r i v e r pressures t o the three-dimensional motion of the d r i v e r gas as i t passes through the opening diaphragm. Using the r e l a t i o n s (7.1) t o (7.3) and standard shock tube theory, one can then p r e d i c t the shock speed from the fr e e parameters, the t e s t and d r i v e r gas f i l l i n g p r essures, and the discharge voltage V . In order to determine the length of the shock-heated gas sl u g contained between the shock f r o n t and the contact s u r f a c e , we have used a shock r e f l e c t i o n technique. This technique i d e n t i f i e s not only the shock f r o n t i n smear camera photographs of the gas flow, but also the"contact surface since the 121 r e f l e c t e d shock i s r e f r a c t e d on passing through t h i s s u r f a c e . We have measured the length of the shock-heated gas s l u g , 1 , at a p o s i t i o n i n the shock tube 120 cm from the diaphragm f o r a c a p a c i t o r voltage of 10 KV and f o r argon t e s t gas pressures, p^, i n the range 0.5 Torr < p^ < 15 To r r . In a l l cases we have found that 1 i s l e s s than the v a l u e , l t n , p r e d i c t e d by i d e a l shock tube theory. Furthermore, we have found from our study of 1 and i t s rate of increase that under the above c o n d i t i o n s two m regimes of gas flow can be d i s t i n g u i s h e d by whether p^ i s l e s s than or greater than 4 Torr. A simple model has been formulated to e x p l a i n our r e s u l t s . For a l l p^, we f i n d that t e s t gas i n i t i a l l y i n the f i r s t 60 cm of the shock tube mixes with the d r i v e r gas and so i s l o s t . For < 4 T o r r , where d± m/dt < d l ^ / d t , a f u r t h e r l o s s mechanism, probably a viscous boundary l a y e r , removes some of the shock-heated gas. For p^ > 4 T o r r , where dl^/dt....> d l ^ / d t , the gas flow appears to be dominated by the r a r e f a c t i o n wave r e f l e c t e d from the d r i v e r s e c t i o n . This wave overtakes and reta r d s the contact s u r f a c e . Whereas the removal of shock-heated gas does not a l t e r .the p r o p e r t i e s of the r e t a i n e d gas, the presence of the r a r e f a c t i o n wave does a l t e r the shock-heated gas p r o p e r t i e s and so cannot be t o l e r a t e d . A knowledge of the l i m i t a t i o n s of a shock tube i n t h i s respect i s e s s e n t i a l . Our i n v e s t i g a t i o n s show tha t the length of the shock-heated gas sample, under the above c o n d i t i o n s , v a r i e s from 5 cm f o r a Mach 20 shock i n argon at 0.5 Torr to 12 cm f o r a Mach 12 shock i n argon at 3 T o r r , at a p o s i t i o n 120 cm from the diaphragm. 122 The p r e s s u r e , p 2 , behind the shock f r o n t was studied w i t h a p i e z o e l e c t r i c pressure probe and was found to agree w i t h the p r e d i c t i o n s of Rankine-Hugoniot theory. A l s o , the tempera-t u r e , T 2, of the shock-heated gas was determined from the sound speed, a 2 , which was measured by applying a t i m e - o f - f l i g h t technique to a small pressure disturbance introduced i n t o the hot gas. Behind a Mach 13 shock i n a r g o n . i n i t i a l l y a p^ = 1 Torr ( i n which case the degree of i o n i z a t i o n i s c a l c u l a t e d to be 71), the temperature measured i n t h i s way agreed very w e l l w i t h the temperature c a l c u l a t e d from the shock speed u s i n g the theory of Chapter 2. I t appears, then, from t h i s study of p 2 and T 2, that the shock-heated gas p r o p e r t i e s conform w e l l to the e q u i l i b r i u m p r e d i c t i o n s of the Rankine-Hugoniot theory i n the regime s t u d i e d . In summary, our i n v e s t i g a t i o n s show that the Smy shock tube i s an e x c e l l e n t source of plasma of p r e d i c t a b l e p r o p e r t i e s . 7 . 2 Suggestions f o r Future Work Our i n v e s t i g a t i o n s of the separation of shock f r o n t and contact surface showed that the r a r e f a c t i o n wave r e f l e c t e d from the d r i v e r s e c t i o n dominated the gas flow f o r p^ > 4 T o r r , at a bank voltage of 10 KV and at a p o s i t i o n 120 cm from the diaphragm. To extend the u s e f u l pressure range of the shock tube above 4 T o r r , an attempt should be made to lengthen the d r i v e r s e c t i o n . (Our d r i v e r s e c t i o n was 7.6 cm long.) This would a l s o enable a l a r g e r sample of shock-heated gas to be obtained by moving the observation s t a t i o n f a r t h e r away from the diaphragm. 123 In order to make f u l l use of the cap-acitor bank energy, an attempt to e l i m i n a t e the e x t e r n a l spark gap swi t c h should be made. The d r i v e r s e c t i o n would have to be modified to hold o f f the d e s i r e d voltages and to include a " t r i g g e r " f a c i l i t y . I t may be p o s s i b l e to reduce energy losses from the hot d r i v e r gas by usi n g e l e c t r o d e m a t e r i a l s of low thermal con-. d u c t i v i t y (e.g. s t a i n l e s s s t e e l ) and by l i n i n g the d r i v e r s e c t i o n (27 2 8 1 w i t h a heat r e s i s t a n t m a t e r i a l (e.g. Lexan^ ' J) which i s not e a s i l y evaporated. A l s o , d i l u t i o n of the d r i v e r gas wit h high molecular weight m a t e r i a l s evaporated from w a l l s and elect r o d e s . i s u n desirable because t h i s reduces the d r i v e r gas sound speed. In the present shock tube design, the heated d r i v e r gas can c o o l appreciably during the i n t e r v a l ( »v 70 ysec) bet\\?een the discharge and the diaphragm rupture. To overcome t h i s problem, and so increase the speed of shocks a v a i l a b l e , two c a p a c i t o r discharges could be used, the f i r s t to open the diaphragm, and the second to heat the d r i v e r gas j u s t as the diaphragm begins to open. The second c a p a c i t o r bank should discharge q u i c k l y so that the d r i v e r gas i s heated before i t escapes. 124 BIBLIOGRAPHY 1. A. C. Kolb, P h y s i c a l Review107, 345 (1957 2. A. G. Gaydon and I . R. H u r l e , The Shock Tube i n High  Temperature Chemical P h y s i c s , CTTapman and H a l l , London (1963) 3. H. Muntenbruch, p r i v a t e communication 4. H. Muntenbruch, IPP 3/58 (1967) I n s t i t u t f u r Plasmaphysik Garching b e i Munchen 5. P. R. Smy, Nature 193, 969 (1962) 6. P. R. Smy, Review of S c i e n t i f i c Instruments 3_6 , 1334 (1965) 7. J . N. Bradley, Shock Waves i n Chemistry and P h y s i c s , Methuen, London (1962) 8. Ya. B. Zel'dovich and Yu. P. R a i z e r , Physics of Shock Waves  and High-Temperature Hydrodynamic Phenomena, Academic P r e s s , New York (1966) ~ 9. E . L . R e s l e r , S. L i n and A. Kantrowitz, J o u r n a l of App l i e d Physics 23_, 1390 (1952) 10. 0. Laporte, "High Temperature Shock Waves" i n Combustion  and P r o p u l s i o n T h i r d Agard Colloquium, Pergamon Press (T958) 11. A. B. Cambel, D. P. Duclos, and T. P. Anderson, Real Gases, Academic Pr e s s , New York (1963) 12. H. Nett, IPP 3/43 (1966) 13. J.W. Bond, P h y s i c a l Review 105, 1683 (1957) 14. H. Petschek and S. Byron, Annals of Physics 1, 270 (1957) 15. H. Wong and D. Bershader, J o u r n a l of F l u i d Mechanics 26, 459 (1966) 16. R. M. Hobson, Eighth I n t e r n a t i o n a l Conference on Phenomena i n Ionized Gases, A Survey of Phenomena i n Ionized Gases: I n v i t e d Papers Vienna (19671 17. K. E. Harwell and R. G. Jahn, Physics of F l u i d s 7_, 214 (1965) 18. V. H. Blackman and G. B. F. N i b l e t t . i n Fundamental Data  Obtained from Shock Tube Experiments, e d i t e d T y A. F e r r i , Pergamon Press (1961) 125 19. E . J . Morgan and R. D. Morrison, Physics of F l u i d s 8_, 1608 (1965) 20. S. S. Medley, M.'Sc. T h e s i s , Department of P h y s i c s , U n i v e r s i t y of B r i t i s h Columbia (1965) 21. S. L. Leonard, i n Plasma Diag n o s t i c Techniques, e d i t e d by R. H. Huddlestone and S. L. Leonard, Academic Press New York (1965) 22. C. C. Daughney, M.Sc. T h e s i s , Department of P h y s i c s , U n i v e r s i t y of B r i t i s h Columbia (1963) 23. G. A. Theophanis, Review of S c i e n t i f i c Instruments 31, 427 (1960) 24. D. R. White, J o u r n a l of F l u i d Mechanics 4, 585 (1958) 25. C. J . S. M. Simpson, T. D. R. Chandler, and K. B. Bridgman, Physics of F l u i d s 10, 1894 (1967) 26. J . C. Camm and P. H. Rose, Physics of F l u i d s 6, 663 (1963) 27. T. C. Peng and D. L. L i q u o r n i k , Physics of F l u i d s 8_, 693 (1967) 28. T. C. Peng and D. L. L i q u o r n i k , Aerospace Corporation Report TR66-53 (1966) . 29. W. Roth and P. Gloersen, J o u r n a l of Chemical Physics 29, 820 (1958) 30. C. A. Ford and I . G l a s s , J o u r n a l of the A e r o n a u t i c a l Sciences 2_3, 189 (1956) 31. R. E. Duff, Physics of F l u i d s 2, 207 (1959) 32. W. J . Hooker, Physics of F l u i d s 4, 1451 (1961) 33. J . H. Williamson, B r i t i s h J o u r n a l of A p p l i e d Physics 1_8, 317 (1967) 34. B. Ahlborn, Physics of F l u i d s 9, 1873 (1966) 35. B. Ahlborn and M. S a l v a t , Z e i t s c h r i f t f u r Naturforschung 22a, 260 (1967) 36. A. Roshko, Physics of F l u i d s 3, 835 (1960) 37. G. F. Anderson, J o u r n a l of the Aerospace Sciences 26, 184 (1959) 126 38. H. M i r e l s , Physics of F l u i d s 6, 1201 (1963) 39. W. Katsaros, IPP 1/46 (1966) 40. D. H. Edwards, J o u r n a l of S c i e n t i f i c Instruments 35, 346 (1958) 41. D. H. Edwards, L. Davies, and T. R. Lawrence, J o u r n a l of S c i e n t i f i c Instruments 4_1, 609 (1964) 42. M. 0. Stern and E. N. Dacus, Review of S c i e n t i f i c Instruments 3_2, 140 (1961) 43. T. G. Jones and G. C. V l a s e s , Review of S c i e n t i f i c Instruments 3£, 1038 (1967) 44. K. Buc h l , Z e i t s c h r i f t f u r Naturforschung 19a, 690 (1964) 45. K, Buchl, IPP 1/11 (1963) 46. R. M. Measures, Ph.D. T h e s i s , Imperial College of Science and Technology, U n i v e r s i t y of London (1964) 47. I . R. Jones, Review of S c i e n t i f i c Instruments 3_7, 1059 (1966) 48. I . R. Jones, Aerospace Corporation Report TDR-594 (1208-01) TR3 (1961) 49. A. W. Bla c k s t o c k , H. R. K r a t z , and M. B. Feeney, Review of S c i e n t i f i c Instruments 3_5, 105 (1964) 50. K. W. Ragland and R. E. C u l l e n , Review of S c i e n t i f i c Instruments 3_8 , 740 (1967) 51. E. F. Greene and J . P. Toennies, Chemical Reactions i n  Shock Waves, Edward A r n o l d , London (1964) 52. 0. Laporte and E. B. Turner, i n Fundamental Data Obtained  from Shock Tube Experiments, see reference 18 53. R. G. Fowler, J . S. G o l d s t e i n , and B. E. C l o t f e l t e r , P h y s i c a l Review 82, 879 (1951) 127 APPENDIX - PRESSURE PROBES The pressure probes used i n the i n v e s t i g a t i o n s reported f40 411 i n t h i s t h e s i s are of the bar type developed by Edwards^ ' ' and Stern and D a c u s ^ ^ . Many subsequent improvements and m o d i f i c a t i o n s to t h i s b a s i c probe design and to the c a l i b r a t i o n techniques have been r e p o r t e d ^ * ^ ' ^ 50)^ A n e x c e l l e n t review of the pressure bar technique has been given by J o n e s . Measures has i n v e s t i g a t e d the p r o p e r t i e s of these probes i n great d e t a i l . The b a s i c design i s i l l u s t r a t e d i n Figure A - l . The pressure to be measured i s a p p l i e d normally t o the free end of the f r o n t rod, A. A s t r e s s wave propagates along t h i s rod to a p i e z o e l e c t r i c element B and then i n t o the rear rod, C. E l e c t r i c a l connections are made to B i n a s u i t a b l e manner and a s i g n a l , which i s p r o p o r t i o n a l to the appl i e d f o r c e , can be conveniently observed on an o s c i l l o s c o p e . This type of gauge has a number of features which make i t u s e f u l i n plasma physics experiments. The s e n s i t i v e element A p p l i e d Pressure Front Rod Piezo-e l e c t r i c Element Rear Rod Figure A - l Basic Features of P i e z o e l e c t r i c Pressure Probe 128 and the a s s o c i a t e d e l e c t r i c a l connections can be removed out of the region occupied by the plasma. In t h i s way, the e f f e c t on the element of thermally-induced s t r e s s e s can be avoided. A l s o , the c i r c u i t r y can then be conveniently s h i e l d e d against e l e c t r o -magnetic f i e l d s a s s o c i a t e d w i t h discharge-produced plasmas. In cases where i t i s d i f f i c u l t to exclude r a d i a t e d f i e l d s completely, the measurement of the pressure can be delayed a c o u s t i c a l l y u n t i l a f t e r the r a d i a t e d f i e l d s have decayed by choosing a s u i t a b l e length f o r rod A. The delay i s t d e i - V c l - ( A * 1 } where c^ i s the s t r e s s wave speed i n the f r o n t rod of length 1^. Good s p a t i a l r e s o l u t i o n can be obtained, w i t h a s a c r i f i c e i n s e n s i t i v i t y , by using a f r o n t rod of s u f f i c i e n t l y small diameter. The observation time depends on the length of the rear rod. The s t r e s s wave, a f t e r passing through the p i e z o e l e c t r i c element, enters the rear rod. On r e f l e c t i o n from the free end, i t returns and again a c t i v a t e s the p i e z o e l e c t r i c element. I f the rear rod has length 1 ^ and a s t r e s s wave propagation speed of Cj, the observation time i s t o b s 2 I 3 / C 3 (A.2) and can be of any d e s i r e d d u r a t i o n by s u i t a b l e choice of 1^. The r e a r rod serves another purpose i n mechanically loading the s e n s i t i v e element so t h a t o s c i l l a t i o n s of the element at i t s n a t u r a l frequencies are suppressed. In c o n s t r u c t i n g such a prcbe, some co n s i d e r a t i o n s must be borne i n mind. The tra n s m i s s i o n of the s t r e s s wave between the s e c t i o n s of the composite probe of Figure A - l depends on the 129 r e l a t i v e a c o u s t i c impedances of the adjacent" s e c t i o n s . The a c o u s t i c impedance i s defined as Z = fc (A. 3) where j> = mass den s i t y c = propagation speed \ I f Z j = Z 2 across the i n t e r f a c e between media 1 and 2, 100% t r a n s m i s s i o n of the s t r e s s wave w i l l r e s u l t . Considerable care. . must be taken i n glueing the s e c t i o n s of such a probe together. Even i f the s e c t i o n s themselves are a c o u s t i c a l l y matched, the bonding agent may cause r e f l e c t i o n s which could d i s t o r t the pressure s i g n a l under observation. J o n e s ^ ^ has shown that the glue need not be a c o u s t i c a l l y matched to the r e s t of the probe provided the t h i c k n e s s , d, of the l a y e r of glue i s small compared to the probe r a d i u s , r . For a t y p i c a l epoxy r e s i n , he shows that one must have d < 0.06r fo r 99% t r a n s m i s s i o n . Care must be taken to ensure that there are no a i r bubbles i n the g l u e , however, since even extremely t h i n l a y e r s of a i r can cause severe r e f l e c t i o n s and d i s t o r t i o n of the s i g n a l . In designing a probe, one must a l s o consider the frequency spectrum of the pressure v a r i a t i o n s to be measured. On propagating through the probe f r o n t rod, pressure pulses are d i s p e r s e d ; that i s , d i f f e r e n t frequency components propagate wi t h d i f f e r e n t speeds. The s o l u t i o n of the appropriate e l a s t i c equations f o r the d i s p e r s i o n r e l a t i o n s i s a h i g h l y complex problem. However, i t can be shown to a f a i r degree of accuracy 130 that i f Xtn,n ( A * 4 ) where * m ^ n i s the minimum wavelength present i n the pressure spectrum, then the pulse w i l l propagate u n d i s t o r t e d with speed c o = ( E / p ) 1 / 2 where E i s Young's modulus f o r the m a t e r i a l of the rod. (See J o n e s f o r references.) One should therefore use a rod of small r a d i u s . A more d e t a i l e d a n a l y s i s s h o w s that the s i g n a l r i s e time of a probe subjected to a pressure step i s given by . (/ t - M »*(•$-)' ( i r ) • i . » r ( A . 5 ) where V = Poisson's r a t i o f o r the f r o n t rod 1^ = length of the f r o n t rod (The r i s e time i s defined as the time r e q u i r e d f o r the s i g n a l to r i s e from 10% to 90% of i t s f i n a l value.) Fast r i s e - t i m e s can th e r e f o r e be obtained f o r small V , small r and large C q. The r i s e - t i m e cannot, however, be l e s s than the t r a n s i t time of the pressure pulse through the p i e z o e l e c t r i c element. Equation (A.5) holds only f o r i n f i n i t e l y t h i n sensing elements. J o n e s ^ ^ has obtained a r i s e - t i m e of 0.54 ysec using a 10 cm b e r y l l i u m rod (small 1> , large c ) 5 mm i n diameter and a PZT-4 p i e z o e l e c t r i c ceramic element 0.5 mm t h i c k . A pressure probe must be c a r e f u l l y mounted i n order to allow free motion of the surface of the f r o n t r o d ^ ^ ' 4 ^ . F a i l u r e to do so leads t o d i s t o r t e d s i g n a l s and long r i s e time. Also the bar must be protected from l a t e r a l s t r e s s e s . 131 We have constructed pressure probes-using fused quartz rods of 1 mm diameter and PZT-4 ceramic d i s c s 0.2 mm t h i c k , With f r o n t and r e a r rods 22 cm and 10 cm long r e s p e c t i v e l y , the delay time, r i s e - t i m e , and observation time were 40 ysec, 0.8 ysec, and 37 ysec, r e s p e c t i v e l y . The c o n s t r u c t i o n of the probe and i t s mount i s shown i n d e t a i l i n Figure A-2 and Figure 3-7. The probe f i t t e d l o o s e l y i n s i d e the p l a s t i c tubing which was i n s i d e the Corex p r o t e c t i n g s h i e l d ( k i n d l y donated by the Corning Glass Works). The procedure f o r c o n s t r u c t i n g the probes was as f o l l o w s . The quartz rods were cut to length and then t h e i r ends were c a r e f u l l y ground f l a t and perpendicular to the axes of the rods. The i n s u l a t i o n was removed from the ends of #40 enameled w i r e s . Two or three turns of the bare wire were wrapped around the rods near the ends which were t o be glued to the p i e z o e l e c t r i c d i s c . A t h i n f i l m of conducting p a i n t was then a p p l i e d to the ends of the rods and over the small c o i l s of w i r e . While the p a i n t was s t i l l s o f t , the rods were mounted i n a specially-made v i s e , the PZT-4 d i s c was i n s e r t e d between the ends of the two rods, and the rods were then clamped t i g h t l y against the d i s c . When the p a i n t had d r i e d , the d i s c and rod ends were covered w i t h a small amount of epoxy cement to bond the pieces f i r m l y together. Before the glue hardened, the wires were t w i s t e d t i g h t l y together. Figure A-3 shows s i g n a l s obtained with these probes mounted perpendicular to the axis of the shock tube. / / B r a s s Tube B r a s s O - r i n g S e a l C o v a r S e a l BNC C o n n e c t o r S e t - S c r e w f o r A d j u s t i n g P r o b e B r a s s Tube 4-I C o r e x Tube T w i s t e d # 4 0 ' W i r e L e a d s i i S o f t P l a s t i c T ube F r o n t Q u a r t z Rod PZT-4 P i e z o -e l e c t r i c C e r a m i c D i s c / Rear* Q u a r t z Rod F i g u r e A-2 C o n s t r u c t i o n D e t a i l s o f P r e s s u r e P r o b e H o u s i n g t R e f l e c t i o n from Free End of Rear Rod (a) Mach 9 Shock i n Argon at 10 Torr (100 mV/div, 5 u.sec/div) (b) Mach 13 Shock i n Argon at 1 Torr (10 mV/div, 2 jisec/div) t "Sound" Pulse ( i ) Without "Sound" ( i i ) With "Sound" (c) I l l u s t r a t i n g Use of Pressure Probe i n Sound Speed Measurements Behind Mach 13 Shock i n Argon at 1 Torr (10 mV/div, 2 usec/div) Figure A-3 T y p i c a l Pressure Probe Si g n a l s 

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